A quadratic is a function in the form f(x) = ax²+ bx + c
The graph is a parabola
Quadratic 1: x -4x -21
The Roots refer to the values of x that make f(x) = 0. They're also called x-intercepts and solutions. We'll mainly use the term (root) in this chapter, but the other terms are just as common.
Here we can just factor to find the roots:
x² - 4x -21 = 0
(x-7) (x+3)= 0
x = 7, -3
The roots are 7 and -3. Graphically, this means the quadratic crosses the x-axis at x=7 and x = -3
The Sum and Product of the Roots
We already found the roots, so their sum is just 7 + (-3) = 4 and their product is just 7 x -3 = -21
Given the quadratic: y = ax² + bx + c, the sum of the roots is equal to -b/a and the product is equal to c/a. In our example, a =1, b = -4, c = -21.
Sum -b/a = -4/1 = 4
Product = c/a = -21/1 = -21
The vertex is the mid-point of a parabola.
The X-coordinate is always the mid-point of the two roots, which can be found by averaging them.
Because the roots are 7 and -3, the vertex is at x = 7 + (-3)/2 = 2. When x = 2, f(x) = (2)^2 - 4(2) - 21 = -25.
Therefore the vertex is at (2,-25). The minimum and maximum is always the vertex. Minimum of -25.
Vertex Form: *y = a(x-h)^2 + k. *
To get a quadratic function into vertex form, we have to complete the square.
y = x² - 4x -21
The middle term is 4. The first step is to divide it by 2 to get -2. Then write the following.
y = (x-2)^2 -21
The Second step is to take that -2 and square it. We get 4.
y = (x-2)^2 -21 -4 ----> * y = (x-2)^2 - 25*
If a quadratic is in the form ax^2 + bx + c, then the discriminant is equal to b^2 -4ac.
f(x) = x^2 -4x -21.
Discriminant = b^2 -4ac = (-4)^2 -4(1) (-21) = 100
What matters is the sign of the discriminant. We care its positive.
When Discriminant > 0, there are two real solutions
When D = 0, there is one real solution
When D<0, there are no real solution
The Quadratic Formula
If we must find the roots, one way you can do so is with the quadratic formula.
x = -b ± √b² -4ac/2a
f(x) = x² -4x -21
The roots or solutions are
x = - (-4) ± √(-4)² -4(1) (-21)/2(1)
= 4 ±√100/2 = 4 ± √100/2
=4 ± 10/2 = 7 or -3
Notice the discriminant, b² -4ac > 0, the "±" takes effect and we end up with two different roots. When b² -4ac = 0, the "±" for not have an effect because we are essentially adding and subtracting 0. When b² -4ac < 0,we're taking the square root of a negative number, which is undefined and gives us no real roots.
When exponential growth happens it can be modeled as:
y = ax⁺
where y is the final amount after t time intervals, a is the initial amount, and x is the rate that we multiply by. So if we started off with 100 ants, our equation would be
y = 100(2)⁺
where t is the number of months that have gone by. And if our bank account started off with $200, our equation would look like.
y = 200(1.05)⁺
where t is the number of years.
The only difference between exponential decay and growth is that the rate, x, is less than one. So in the case of radioactive decay the equation may look like
y = 400(0.6)⁺
where y is the final mass, 400 is the initial amount, and t is the number of years that have gone by.
Both exponential decay and linear decay are examples of a negative association between things. As one thing increase, the other decreases
When the data points are close to forming a smooth line, there is a strong association.
A positive association happens when one thing increase and the other does as well.
Example 1: Jacob got 50% of the questions correct on a 30-question test and 90% on a 50 question test. What percent of all questions did jacob get correct?
1) First, let's fin the total number of questions he got correct:
50% x 30 = 1/2 x 30 = 15
90% - 50 = 9/10 x 50 = 45
15 + 45 = 60 questions correct out of a total of 30 + 50 = 80 questions. 60/80 = 3/4 = 75%
Which of the following is closet to the percent of recorded parking violations committed by trucks?
Truck, Parking = 17
Car, Parking = 26
Total, Parking = 43
17/43 = 40%
If the data were used to estimate driving violation information about 2,000 total violations in a certain state, which of the following is the best estimate of the number of speeding violations committed by cars in the state.
Car Speeding Violations/Total Parking Violations = 83/284 x 2000 = 585
Example 4: The price of a dress is increased by 20%, then decreased by 40%, then increased by 25%. The final price is what percent of the original price.
1) Let the original price be "p'.
When "p" is increased by 20% you multiply by 1.20 because it's the original plus 20%, When its decreased by 40%, you multiply by .60 because 60% is what's left after you take away 40%. Our final price is then:
p x 1.20 x .60 x 1.25 = .90p
90% of original.
Never calculate at each step, do it all together.
Example 5: Jonas has a savings account that earns 3 percent interest compounded annually. Her initial deposit was $1000. Which of the following expressions give the values of the account after 10 years?
A percent interest rate compounded annually means he earns 3 percent on the account once a year. Keep in mind that this isn't just 3% on the original amount of $1000. This is 3% of whatever's in the account at the time, including any interest that he's already earned in previous years. This is the meaning of compound interest. So if we're in year 5, he would earn 3% on the original $1000 and 3% on the total interest deposited in year 1-4.
If we try to calculate the total after each and every year, this problem would take forever, let's take what we learned for example 3.
Year 1 total: 1000(1.03) = 1000(1.03)¹
Year 2 total: 1000(1.03) (1.03) = 1000(1.03) ²
Year 3 total 1000(1.03) (1.03) (1.03) = 1000(1.03)³
Year 4 total: 1000(1.03)(1.03)(1.03)(1.03) = 1000(1.03)⁴
Each year is an increase of 3% so it's just 1.03 times whatever value was last year. Think of the price of the dress being increased by 3% ten times.
Year 10 total is 1000(1.03)¹⁰
Most of compound interest questions can be modeled by the equation A = P(1 + r)ⁿ, where A is the total amount accumulated, P is the principal or initial amount, r is the interest rate, and t is the number of interest is received.
Example 5: Jay puts an initial deposit of $400 into bank account that earns 5 percent interest each year, compounded semiannually. Which of the following equations gives the total dollar amount, A, in the account after t years.
The interest is compounded semiannually which means twice a year. So interest is received 2t times. However, we don't receive a full 5% each time interest is received. The 5% interest rate is a yearly figure. We have to divide it by 2 to get the semiannually rate: 2.5%.
D) A= 400 (1.025)²ⁿ
Note that semiannual compounding is better than annual compounding. WITH ANNUAL COMPOUNDING YOU JUST GET 5 percent on the initial amount after on year. That's like 2.5% on the initial amount and then another 2.5% on the initial amount. But with semiannual compounding, you get 2.5% on the initial amount and then you get 2.5% on the mid-year amount, which is greater than the initial amount because it included the first interest payment. Because you've already earned interest before the end of the year, you get a little extra.
If the interest is compounded more than once a year, the previous formula can be generalized to: A = P (1 + r/n)ⁿ⁹
Where A is the total amount accumulated, P is the principal or initial amount, r is the interest rate, t is the number of years, and n is the number of times the interest is compounded each year.
While compound interest lets you earn interest on interest you've earned, simple interest means you get the same amount each time. Interest is earned only on the original amount.
Example 6: An investor decided to offer a business owner a $20,000 loan at simple interest of 5% per year. Which of the following functions gives the total amount, A, in dollars, the investor will receive when the loan is repaid after t years?
At a simple interest of 5%, the investor will receive 20,000(0.05) in interest each year. That amount does not change because the 5% always applies to the original $20,000 under simple interest. So after t years, he will receive a total of 20,000(0.05)t in interest.
The amount he will be repaid after t years is then:
A= Original amount + Total interest
= 20,000 + 20,000 (0.05)t
= 20,000(1 + 0.05t)
D) A = 20,000(1 + 0.05t)
For simple interest the formula: A = P(1 + rt)
where A is the total amount accumulated, P is the principal or initial amount, r is the interest rate, and t is the number of times interest is earned (typically years.)
Example : This year, the chickens on a farm laid 30% less eggs than they did last year. If they laid 3,500 eggs this year, how many did they lay last year.
This year = (.70) (last year)
3,500 = (.70) (last year)
5,000 = last year
Percent Change (aka percent increase/decrease) is calculated as follows:
% change = new value - old value/old value x 100
For example if the price of a dress starts out at 80 dollars and rises to 90 dollars, the percent change is:
90 -80/80 x 100 = 12.5%
Percent change is always based on original value.
Example 8: In a particular store, the number of TVs sold the week of Black Friday was 685. The number of TVs sold the following week was 500. TV sales the week following black Friday were what percent less than TV sales the week of Black Friday (round to nearest percent)
500 - 685/685 ≈ -0.27
Example 9: In a particular store, the number of computers sold the week of Black Friday was 470. The number of computers sold the previous week was 320 . Which of the following best approximates the percent increase in computer sales from the previous week to the week of Black Friday?
470-320/320 ≈ 0.47
This time, the week of Black Friday is not the original basis for percent change. We put the difference over the previous week's number, 320.
Example 10: The number of students at a school decreased 20% from 2010 to 2011. If the number students enrolled in 2011 was k, which of the following expresses the number of students enrolled in 2010 in terms of k?
The answer is NOT 1.20k. Percent change is based off of the original value (from 2010) and not the new value. Let x be the number of students in 2010,
x = 1.25k
Therefore, there were 25% more students in 2010 than in 2011.
Example 11: Among 10th graders at a school, 40% if the students are Red Sox fans. Among those Red Sox fans, 20% are also Celtics fans. What percent of the 10th graders at the school are both Red Sox fans and Celtic fans?
We don't know the number of 10th graders at the school so let's suppose it's 100.
Red Sox fans = 40% of 100 = 40
Celtic and Red Sox Fans = 20% of 40 = 8
Answer is then 8/100 = 8%
Imagine we have a triangle. We know that the angle of a triangle is A = 1/2bh
Let's triple the height. What happens to the Area?
The new height is 3h. The area is then:
Anew = 1/2b(3h) = 3(1/2bh) = 3Aold
Example 1: The radius of a circle is increased by 25%. By what percent does the area increase?
1) Let the original area be Aold. If the original radius is r, then the new radius is 1.25r.
Anew = π(1.25r)² = (1.25)²(πr²) = 1.5625(πr²) = 1.5625 A old
The area increases by 56.25%
The idea is to get a number in front of the old formula. In the previous example, that number turned out to be 1.5625. Also note that the 1.25r was wrapped in parentheses so that the whole thing gets squared. It would've been incorrect to have Anew = π(1.25)r² because we wouldn't be squaring the new radius.
Example 2: The length of a rectangle is increased by 20%. The width is decreased by 20%. Which of the following accurately describes the change in the area of the rectangle?
Originally, A= lw
Anew = (1.20l) (.80w) = 0.96lw = 0.96Aold
The are has decreases by 4%.
Example 3: The force of attraction between two particles can be determined by the formula above, in which F is the force between them, r is the distance between them and q₁ and q₂ are the charges of the two particles. If the distance between two charged particles is doubled, the resulting force of attraction is what fraction of the original force?
Fnew = 9q₁q₂/(2r)² = (½)² (9q₁q₂/r² ) = 1/4(9q₁q₂/r² ) = 1/4Fold
B)1/4 Notice how we do not let constraints like the "9" in the formula affect the result. In getting a number out front, students often make the mistake of mixing that number up with numbers that were originally in the formula.
Example 4: The volume of a cube is tripled. The length of each side must have been increased by approximately what percent?
Now we have to solve backwards. Keep in mind that the volume of a cube is V = s³, where s is the length of each side. Increase each side by some factor and rearrange the terms to extract a number. Only this time, we have to use x.
Vnew = (xs)³
Vnew = x³s³ = x³Vold
Notice how we were still able to extract something out in front, x³. That x³ must be equal to 3 if the new volume is to be tripled the old volume.
x³ = 3
x = 3√3 ≈ 1.44
Each side must have been increase by 44%
A system of equations refers to 2 or more equations that deal with the same set of variables.
-5x + y = -7
-3x + 2y = -12
There are two main ways of solving systems of equations: substitution and elimination.
Substitution is all about isolating one variable, either x or y, in the fastest way possible.
Taking the example above, we can see that it's easier to isolate y in the first equation because it has no coefficient. Adding 5x to both sides we get
y = 5x -7
We can then substitute y in the second equation with 5x - 7 and solve from there.
-3x -2(5x-7) =-12
-3x - 10x + 14 = -12
-13x = -26
x = 2
Substituting x =2 back into y = 5x -7, y = 5(2) - 7 = 3
The solution is x =2, y =. which can be denoted as (2,3)
Elimination is about getting the same coefficients for one variable across the two equations so that you can add or subtract the equations, thereby eliminating that variable.
Using the same example, we can multiply the first equation by 2 so that y's have the same coefficient (we don't worry about the sign because we can add or subtract the equations)
-10x + 2y = -14
-3x -2y = -12
To eliminate y, we add the equations
-10x + 2y = -14
-3x - 2y = -12
-13x = -26
Now we can see that x =2. This result can be used in either of the original equations to solve for y. We'll pick the first equation.
-10(2) + 2y = -14
-20 + 2y = -14
2y = 6
y = 3
And finally, we get the same solution we got using x =2, y= 3
When solving systems of equations you can use either method, but one of them will typically be faster. If you see a variable with no coefficient, like in -5x + y = -7 above, substitution is likely the best route. If you see matching coefficients or you see that it's easy to get matching coefficient, elimination is likely the best route. The example above was simple enough for both methods to work well.
A system of equations has no solutions when the same equation is set to a different constant.
3x + 2y = 5
3x + 2y = -4
The equations contradict each other.
3x + 2y = 5
6x + 4y = -8
also has no solution, because the second equation can be divided by 2 to get the original equations.
-ax - 12y = 15
4x + 3y = -2
If the system of equation above has no solution, what is the value of a?
We must get the coefficients to match so that we can compare the two equations. To do that, we multiply the second equation by -4:
-ax -12y = 15
-16x - 12y = 8
See how the -12's match now. Now let's compare. If *a = 16*, then we get our two contradicting equations with no solution. One is set to 15 and the other is set to 8.
A system of equations has infinite solutions when both equations are essentially the same.
3x + 2y = 5
3x + 2y = 5
(1,1), (3,-2), (5,-5) are all solutions the systems above.
6x + 4y = 10
3x + 2y = 5
also has an infinite number of solutions. The first equation can be divided by 2 to get the sane equation.
3x - 5y = 8
mx - ny = 32
In the system of equations above, m and n are constants. If the system has infinitely many solutions, what is the value of m + n.
Both equations need to be the same for there to be an infinite number of solutions. We multiply the first equations by 4 to get the right hand sides to match.
12x - 20y = 32
mx - ny = 32
Now we can clearly see that m =12 and n = 20. Therefore m + n = 32.
You will most definitely run into a question that asks you to translate a situation into a system of equations. Here's a classic example.
A group of 30 students order launch from a restaurant. Each student gets either a burger or a salad. The price of a burger is $5 and the price of a salad is $6. If the group spent a total of $162, how many students ordered burgers?
Let x be the number of students who ordered burgers and y be the number of who ordered salads. We can then make two equations.
x + y = 30
5x + 6y = 162
We'll use elimination to solve this equation. Multiply the first equation by 6 and subtract:
6x + 6y = 180
5x + 6y = 162
x = 18.18 students got burgers.
*More complex Systems*
You might encounter systems of equations that are a bit more complicated than the standards one's you've seen above. For these, substitution and some equation manipulation will do the trick.
y + 3x = 0
x² + 2y² = 76
If (x,y) is a solution to the system of equations above and y > 0, what is the value of y?
In the first equation, we isolate y to get y = -3x. Plugging this into the second equations,
x² + 2(-3x)² = 76
x² + 2(9x²) = 76
x² + 18x² = 76
19x² = 76
x² = 4
x = ± 2
If x =2, then y = -3(2) = -6. If x = -2, then y = -3(-2) = 6. Because y > 0, y = 6
xy + 2y = 1
(1/x + 2)² + (1/x +2) - 6 = 0
If (x,y) is a solution to the equation above, what is a possible value for |y|?
Notice the (x+2)'s lying around. This is a hint that there might be a clever substitution somewhere, especially for a problem as complicated as this one. Isolating y in the first equation,
xy + 2y = 2
y(x + 2) = 2
y = 2/x+ 2
From here, y/2 = 1/x+2. So you can substitute 1/x+2 in the second equation with y/2. As you do these tougher equations, you must keep an eye out for any simplifying manipulations such as this.
Substituting we get,
(1/x+2)² + (1/x+2) - 6 = 0
(y/2)² + (y/2) - 6 = 0
y²/4 + y/2 - 6 = 0
y² + 2y - 24 = 0
(y + 6) ( y-4) = 0
Finally, y = -6 or 4. and |y|can be either 6 or 4.
If any xy = 8, xz = 5, and yz = 10, what is the value of xyz.
Multiply all three equations. Multiply the left sides, and multiply the right sides. The result is:
x²y²z² = 400
Square root both sides:
xyz = 20.
Notice how we were able to get the answer without knowing the values of x, y,z.
The solution to a graph is the intersecting parts. Solutions are equal to that number.
Graphs with no solutions don't touch.
Infinite solutions are the same line.
The mean is the average.
The median is the number in the middle.
When there is no middle number you take the average of the two middle numbers.What if the list was 100 numbers long? Take half to get 50. The 50th and 51th would be the ones in the middle and you would average.
The mode is the number that shows up the most often.
The range is the difference between the biggest number and the smallest.
The *Standard Deviation* is a measure of how spread out a list of numbers is. In other words, how much they "deviate" from the mean. The standard deviation is lower when more numbers are closer to the mean. The standard deviation is higher when more numbers area spread out away from the mean. For example, our list:
would have a higher standard deviation than the following list:
because the second list is more tightly clustered around the mean. It turns out that the standard deviation of our list is 2.28 and the standard deviation of the second list is 0.83. It's not necessary to calculate these numbers.!
* Example 1* The sum of three consecutive integers is 72. What is the largest of these three integers?
Let a variable be what you don't know. We on't know any of the three integers. Let the smallest be x.
Our consecutive integers are then: x, x + 1, x + 2
Because they sum to 72, we can make an equation:
x + (x+1) + (x+2) = 72
3x + 3 = 72
3x = 69
x = 23
And because x is the smallest, our three consecutive integers must be
The largest one is 25.
What if we let x be the largest integer? Or three integers would've been
x -2, x-1, x
An our solution would have looked like:
(x-2) + (x-1) + x = 72
3x -3 = 72
3x = 75
x = 25
And because x was set to the largest of the three integers this time, we're already at the answer.
*Example 2:* One number is 3 times another number. If they sum to 44, what is the larger of the two numbers?
In this problem, we want set x to be the smaller of the two numbers. That way, the two numbers can be expressed as
x and 3x
If we let x be the larger of the two, we would have to work with
x and x/3
Setting up our equation:
x + 3x = 44
4x = 44
x = 11
It asks for the larger of the two, so we have to multiply x by 3 to get 33.
*Example 3:* What is a number such that the square of the number is equal to 2.7% of its reciprocal?
Let the number we're looking for be x.
x² = 0.27 × 1/x
Multiply both sides by x to isolate it
x³ = .027
Cube root both sides x = .3
*Example 4* Albert is 7 years older than Henry. In five years, Albert will be twice as old as Henry. How old is Albert now?
Let x be Albert's age now. We could've assigned x to be Henry's age, but as we mentioned earlier, assigning the variable to be what the question is asking for is typically the faster route. Not at this point, some of you might be thinking of assigning another variable to Henry's age. While that would certainly work, it would only add more steps to the solution. Try to stick to one variable unless the equation clearly calls for more.
If Albert is x years old now, then Henry must be x-7 years old.
Five years from now, Albert will be x+5 and Henry will be x-2 years old.
x+5 = 2(x-2)
x + 5 = 2x - 4
x = 9
*Example 5:* Jake can run 60 yards per minute. Amy can run 120 yards per minute for the first ten minutes but then slows down to 20 yards per minutes thereafter. If they start running at the same time, after how many minutes t will both Jake and Amy have run the same distance, assuming t >10?
The problem already gives us a variable to to work with. We want to equate Jake's distance run with Amy's.
Jake's Distance: 60t
Amy's Distance: 120(10) + 20(t-10)
60t = 120(10) + 20(t-10)
60t = 1,200 + 20t - 200
40t = 1,000
t = 25
After 25 minutes they will have run the same distance.
*Example 6:* At a pharmaceutical company, research equipment must be shared among the scientists. There is one microscope for every 4 scientist, one centrifuge, for every 3 scientists, and one freezer for every two scientists. If there is a total of 52 pieces of research equipment at this company, how many scientists are there?
Let x, be the number of scientists. The number of microscopes is x/4, the number of centrifuges is x/3, and the number of freezers is x/2.
x/4 + x/3 + x/2 = 52
Multiply both sides by 12 to get rid of the fractions,
3x + 4x + 6x = 52x12
13x = 624
x = 48
*Example 7* Mark and Kevin own 1/4 and 1/3 of the books on a shelf, respectively. Lori owns the rest of the books. If Kevin owns 9 more books than Mark, how many books does Lori own?
Let x be the total number of books. Mark then has 1/4x books and Kevin has 1/3x books. Kevin owns 9 more than Mark, so
1/3x - 1/4x = 9
Multiply both sides by 12,
4x -3x = 108
x = 108
The total number of books is 108. Mark owns 1/4 x 108 = 27 books and Kevin owns 1/3 x 108 = 36 books. Lori must then own 108 - 27 - 36 = 45 books.
*Example 8:* A group of friends want to split the cost of renting a cabin equally. If each friend pays $130 they will have $10 too much. If each friend pays $120, they will have $50 too little. How much does it cost to rent the cabin?
We have two unknowns in this problem. We'll let the number of people in the group be n and the cost of renting a cabin be c. From the information given, we can come up with two equations (make sure you see the reasoning behind them)
130n - 10 = c
120n + 50 = c
In the first equation, 130n represents the total amount the group pays, but because that's 10 dollars too much, we need to subtract 10 to arrive at the cost of the rent, c. In the second equation, 120n represents the total amount the group pays, but this time it's 50 dollars too little, so we need to add 50 to arrive at c. Substituting c from the first equation into the second, we get:
120n + 50 = 130n -10
-10n = -60
n = 6
So there are 6 friends in the group and
c = 130n - 10 = 130 x 6 - 10 = 770
The cost of renting the cabin is 770.
*Example 9:* Of the 200 jellybeans in a jar, 70% are green and the rest are red. How many green jelly beans must be removed so that 60% of the remaining jellybeans are greens.
The answer is not 20. We first find that there are 7/10 x 200 = 140 green jellybeans. We need to remove x of them so that 60% of what's left is green:
green jellybeans left/total jellybeans left = 60%
140-x/200-x = 6/10
10(140-x) = 6(200-x)
1,400 - 10x = 1,200 - 6x
200 = 4x
x = 50
50 green jelly beans need to removed.
*Example 10* Altogether, David and Robert have 120 baseball cards. Davids gives Robert 1/3 of his cars and then 10 more cards. Robert now has five times as many cards as David. How many cards did Robert have originally?
Solution 1: This question is really tough and tricky. When David gives Robert some of his cards, David loses at the same time Robert gains. We could set a variable for David and another variable for Robert, but that solution is a little messier.
Instead, let's work backwards. If x is the number of cards David ends up with, then Robert ends up with 5x cards. Because there are 120 cards altogether,
x + 5x = 120
6x = 120
x = 20
So David has 20 cards and Robert has 100 cards at the end. Let's rollback another transaction. David had given Robert 10 cards. So before that happened, David must have had 20 + 10 = 30 cards. Rollback another transaction ad we see that David had given a third of his cards away to get down to the 30 that we just calculate. Well if he had given away a third, then the 30 he had left must have represented two-thirds of the cards he had at the start. Let d be the number of cards David had at the star.
So David had 45 cards at the start, which means Robert must have had 120 - 45 = 75 cards at the start.
Solution 2: Let x be the number of cards David starts with and y be the number of cards Robert starts with. He are the equations:
x + y = 120
y + 1/3x + 10 = 5(x - 1/3x -10)
Multiplying the second equation by 3,
x + y = 120
3y + x + 30 = 15x - 5x - 150
Shifting things over,
x + y = 120
9x - 3y - 180
At this point, we can use substitution or elimination. I'm going to use substitution. From the first equation, x = 120 - y. Plugging this into the second equation,
9(120 -y) - 3y = 180
1080 - 12y = 180
-12y = -900
y = 75
While the math section doesn't place a large emphasis on geometry problems, it does focus on algebra, solving equations, and data interpretation from tables and graphs. College Board sorts the question types into three main categories: Heart of Algebra, Passport to Advanced Math, and Problem Solving and Data Analysis (they apparently gave up on the creative naming once they reached the third category).
These three realms describe about 90% of the SAT math questions. The remaining 10% are simply called Additional Topics, and they mainly include geometry, basic trigonometry, and complex numbers.
Let's take a closer look at each of these categories by going over the SAT math topics and skills they test. After a description of each one, you'll see three official sample practice questions from College Board.
Heart of Algebra
SAT math questions in the Heart of Algebra category have to do with linear equations, inequalities, functions, and graphs. Below are the official topics as defined by College Board, followed by a summary of tasks you'll need to be prepared for to tackle these questions and some example problems.
Solving linear equations and linear inequalities (in these expressions, x is a constant or the product of a constant)
Interpreting linear functions
Linear inequality and equation word problems
Graphing linear equations
Linear function word problems
Systems of linear inequalities word problems
Solving systems of linear equations
Summary of Tasks
Use multiple steps to simplify an expression or equation or solve for a variable.
Solve for a variable within functions or systems of inequalities with two variables (usually x and y).
Determine whether a given point is in a solution set or what value would make an expression have no solution.
Select a graph that shows an algebraic equation, or, on the flip side, choose the equation that describes a graph.
Indicate how a graph would be affected by a given change in its equation.
Passport to Advanced Math
While Heart of Algebra questions are focused on linear equations, Passport to Advanced Math questions have to do with nonlinear expressions, or expressions in which a variable is raised to an exponent that's not zero or one. These questions will ask you to work with quadratic equations, exponential expressions, and word problems. Read on for the full list of topics that fall under Passport to Advanced Math, followed by a summary of tasks and three sample SAT questions
Solving quadratic equations
Interpreting nonlinear expressions
Quadratic and exponential word problems
Radicals and rational exponents
Operations with rational expressions and polynomials
Polynomial factors and graphs
Nonlinear equation graphs
Linear and quadratic systems
Structure in expressions
Summary of Tasks
Solve equations by factoring or using other methods to rewrite them in another form.
Add, subtract, multiply, or divide two rational expressions or divide two polynomial expressions and simplify your results.
Select a graph that matches a nonlinear equation or an equation that corresponds to a graph.
Determine the equation of a curve from a description of a graph.
Figure out how a graph would change if its equation changed.
Problem Solving and Data Analysis
This third and final major category includes questions that ask you to work with rates, ratios, percentages, and data from graphs and tables. Read on for the official topics, a summary of tasks, and three sample questions.
Ratios, rates, and proportions
Key features of graphs
Linear and exponential growth
Center, spread, and shape of distributions
Data collection and conclusions
Summary of Tasks
Solve multi-step problems to calculate ratio, rate, percentage, unit rate, or density.
Use a given ratio, rate, percentage, unit rate, or density to solve a multistep problem.
Select an equation that best fits a scatterplot.
Use tables to summarize data, such as probabilities.
Estimate populations based on sample data.
Use statistics to determine mean, median, mode, range, and/or standard deviation.
Evaluate tables, graphs, or text summaries.
Determine the accuracy of a data collection method
Additional Topics in Math
While 90% of your questions will fall into the Heart of Algebra, Passport to Advanced Math, or Problem Solving and Data Analysis categories, the remaining 10% will simply be classified as Additional Topics. These topics include geometry, trigonometry, and problems with complex numbers.
Volume word problems
Right triangle word problems
Congruence and similarity
Right triangle geometry
Angles, arc lengths, and trig functions
Summary of Tasks
Determine volume of a shape.
Apply properties of triangles to determine side length or angle measure.
Apply properties of circles to measure arc length and area.
Solve problems with sine, cosine, and tangent.
For any triangle, the sum of any two sides must be greater than the third.
a + b > c
b + c > a
a + c > b
Example 1: In ∆ABC, AB has a length of 3 and BC has a length of 3. How many integers valuers are possible for the length of AC?
Let the length of AC be x. Based on the rule, we can come up with three equations:
3 + 4 > x
3 + x > 4
4 + x > 3
which simplify to:
7 > x
x > 1
x > -1
Now if x > 1, then it's always going to be greater than -1. In other words, only the first and second matter. Therefore, 1 < x < 7, and there 5 possible integer values of x.
* Isosceles and Equilateral Triangles*
An isosceles triangle is one that has two sides of equal length. The angles opposite those sides are equal.
Equilateral triangles have all equal sides.
*Example 2:* In isosceles triangle, one of the angles has a measure of 50°. What is the degree measure of the greatest possible angle in the triangle?
An isosceles triangle has not only two equal sides, but also two equal angles. There are two possibilities for an isosceles triangle with an angle of 50°. Another angle could be 50°, making a 50-50-80 triangle, or the other two angles could be equal, making a 50-65-65 triangle. Given these two possibilities, 80° is the greatest possible angle in the triangle.
*Example 3:* In the figure above, the triangle ABC is equilateral. What is the value of j + k + l + m + n + o?
Solution 1: There are three smaller triangles within the equilateral one. Each of these triangles has a total degree measure of 180° x 3 = 540°. We need to subtract out <ACB to get what we want.
Because triangle ABC is equilateral, <ACB is 60°. So 540° - 60°= 480°
Right triangles are made up of two legs and the hypotenuse (the side opposite the right angle)
Every right triangle obeys the Pythagorean theorem: a² + b² = c², where a and b are the lengths of the legs and c is the length of the hypotenuse.
The rectangle above has a diagonal length of 20. If the base of the rectangle is twice as long as the height, what is the height.
The diagonal of any rectangle forms two right triangle. Let the height be x and the base be 2x. Using the Pythagorean theorem,
x² + (2x)² = 20²
x² + 4x² = 400
5x² = 400
x² = 80
x = √80 = 4√5
If you take the SAT enough times, what you'll find is that certain right triangles come up repeatedly. For example, the 3-4-5 triangle.
A set of three whole numbers that satisfy the Pythagorean theorem is called a Pythagorean triple. Though not, necessary it'll save you quite a bit of time and improve your accuracy if you learn to recognize the common triples that show up:
Note that the 6-8-10 triangle is just a multiple of the 3-4-5 triangle.
* Special Right Triangles*
You will have to memorize two special right triangle relationships. The first is the 45-45-90:
The best way to think about this triangle is that it's isosceles- the two legs are equal. We let their lengths be x. The hypotenuse, which is always the biggest side in a right triangle, turns out to be √2 times x.
We can prove this relationship using the Pythagorean theorem, where h is the hypotenuse
x² + x² = h²
2x² = h²
√2x² = √h²
x√2 = h
I show you these proofs not because they will be tested on the SAT, but because they illustrate problem-solving concepts you may have to use on the SAT.
The Second is the 30-60-90:
Because 30° is the smallest angle, the side opposite from it is the shortest. Let that side be x. The hypotenuse, the largest side, turns out to be twice x, and the side opposite 60° turns out to be √3 times x.
One common mistake students make is to think that because 60° is twice 30°, the side opposite 60° must be twice as big as the side opposite 30°. That relationship is NOT true. You cannot extrapolate the ration of the sides from the ratio of the angle. Yes, the side opposite 60° is bigger than the side opposite 30°, but it isn't twice as long.
We can prove the 30-60-90 relationship by using an equilateral triangle. Let each side be 2x. (we could use x but you'll see why 2x makes things easier)
Drawing a line down the middle from B to AC creates two 30-60-90 triangles. Because an equilateral triangle is symmetrical, AD is half of 2x, or just x. That's why 2x was used- it avoids an fractions.
To find BD, we use the Pythagorean theorem:
AD² + BD² = AB²
x² + BD² = (2x)²
BD² = 4x² - x²
BD² = 3x²
√BD² = √3x²
Triangle ABD is proof of the 30-60-90 relationship.
What is the area of ∆ACB is shown above.
Using the 45-45-90 triangle relationship, AC = BC = 4/√2 (the hypotenuse is √2) time greater than each leg). The area is then ½(4/√2) (4√2) = 1/2(16/2) = 4
Answer is 4.
In the figure above, AD = DC, <B = 30°, and AB = 10. What is the ratio of AC to CB?
Because AD = DC, ∆ADC is not only isosceles but also 45-45-90 triangle. ∆ADB is a 30-60-90 triangle with a hypotenuse of 10. Using the 30-60-90 relationship, AD is half the hypotenuse, 5, and DB = 5√3. Using the 45-45-90 relationship, AC = 5√2, DC = 5, and CB = DB = 5√3 - 5
AC/CB = 5√2/5√3-5 = *√2/√3 -1*
When two triangles have the same angle measures, their sides are proportional.
Because DE is parallel to AC in the figure above, <BED is equal to <BCA. That makes ∆DBE similar to ∆ABC. In other words, ∆DBE is just a smaller version of ∆ABC. If we draw the two triangles separately and give the sides some arbitrary lengths, we can see this more clearly.
The sides of the big triangle are twice as long as the sides of the smaller one. Even if the lengths change, the ratios will remain the same:
AB/BD = AC/DE = BC/BE
Part 1: In ∆ABC above, DE is parallel to AC, AD = 2, DB = 3, and DE = 6. What is the length of AC?
Part 2: What is the ratio of the area of ∆BDE to the area of ∆BAC?
Part 1 Solution: Because DE and AC are parallel, <BDE is equal to < BAC and <BED is equal to < BCA. Therefore, ∆BDE and ∆BAC are similar. Setting up are ratios,
BD/BA = DE/AC
3/5 = 6/AC
3AC = 30
AC = 10
Part 2 Solution: When two triangles are similar, the ratio of their areas is equal to the square of the ratio of their sides. The ratio of the sides is 3:5. Squaring that ratio, we get the ratio of the areas, 9:25
Answer is: 9/25.
A radian is simply another unit used to measure angles. Just as we have feet and meters, pounds and kilograms, we have degrees and radians.
π radians = 180°
If you've never used radians before, don't be put off by the π. After all, it's just a number. We could've written:
3.14 radians ≈ 180°
Instead, but everything is typically expressed in terms of π when we're working with radians. Furthermore, 3.14 is only an approximation. So, given the conversion factor above, how would we convert 45° to radians?
45° x πradians/180° = π/4 radians
Notice that the degree units (represented by the little circles) cancel out just as they should in any conversion problem. Now how would we convert 3π/2 radians to degrees? Flip the conversion factor.
3π/2 radians x 180°/πradians = 270°
You might be wondering why we even need radians. Why not just stick with degrees? Is this another difference between the U.S. and the rest of the world, like it is with feet and meters? Nope. As we'll see in the chapter on circles, some calculations are much easier when angles are expressed in radians.
In the xy-plane above, line m passes through the origin and has a slope of √3. If point A lies on line m and point B lies on the x-axis as shown, what is the measure, in radians, of angle AOB?
We can draw a line down from A to the x-axis to make a right triangle. Because the slope is √3, the ratio of the height of this triangle to its base is always √3 to 1 (rise over run)
This right triangle should look familiar to you. It's the 30-60-90 triangle. Angle AOB is opposite the √3, so its measure is 60°. Converting that to radians:
60° x π/180 = π/3
*Exterior Angle Theorem*
An exterior angle is formed when any side of a triangle is extended. In the triangle below, x° designates an exterior angle.
An exterior angle is always equal to the sum of the two angles in the triangle furthest from it. In this case. x = a + b
When two lines are parallel, the following are true:
Vertical angles are equal (eg: <1 and <4)
Alternate Interior angles are equal (<4 = <5, and <3 = <6)
Corresponding angles are equal(<1 and < 5)
Same side interior angles are supplementary (<3 + <5 = 180 degrees)
You just need to know that when two parallel lines are cut by another line, there are two sets of equal angles:
<1 = <4 = <5 = <8
<2 = <3 = <6 = < 7
For any polygon, the sum of the interior angles is
180 (n-2) where n is the number of sides.
So for an octagon, which has 8 sides, the sum of the interior angles is 180(8-2) = 180 x 6 = 1080 degrees.
A regular polygon is one in which all sides and angles are equal. The polygons shown above are regular. If our octagon were regular, each interior angle would have a measure of 1080 degrees divided by 8 = 135°
The 180(n-2) formula comes from the fact that any polygon can be split up into several triangles by drawing lines from one vertex to the others.
The number of triangles that results from this process is always two less than the number of sides. Because each triangle angles 180°, the sum of the angles within a polygon must be 180°, the sum of the angle within a polygon must be 180°(n-2) where n is the number of sides.
Triangle = 180
Quadrilateral = 360
Pentagon = 540
Hexagon = 720
Two sides of a regular pentagon are extended as shown in the figure above. What is the value of?
The total number of degrees in a pentagon is 180(5-2) = 540°. So each interior angle must be 540° ÷ 5 = 108°. The angles within the triangle formed by the intersecting lines must be 180 - 180 = 72°
So x = 180 - 72 - 72 = 36°
1) Matching coefficients
Example 1: If (x + a)² = x² + 8x +b, what is the value of b?
It's hard to see anything meaningful right away on both sides of the equation. So let's expand the left side first and see.
(x + a)² = (x + a) (x + a) = x² + 2ax + a²
So now we have,
x² + 2ax + a² = x² + 8x + b
We can match up the coefficients.
x² + 2ax + a² = x² + 8x + b
2a = 8
a² = b
Solving the equations, a = 4, b=16
*2. Clearing denominators:*
When you solve an equation like ½x + 1/3x = 10, a likely first step is to get rid of the fractions, which are harder to work with. By multiplying both sides by 6. But where did six come from? 2 times 3. So this is what you're actually doing when you multiply by 6:
½x ×(2×3) + 1/3x ×(2×3) = 10 ( 2 × 3)
1x × (3) + 1x × (2) = 10 (2×3)
3x + 2x = 60
We got rid of the fractions by clearing denominators.
3/x + 5/x+2 = 2
If x is a solution to the equation above and x >0, what is the value of x?
In the same we multiplied by 2 x 3, we can multiply by x(x +2) here.
3/x × x(x+2) + 5/x+2 ×(x+2) = 2 × x(x+2)
3(x+2) + 5x = 2x² + 4x
3x + 6 + 5x = 2x² + 4x
0 = 2x² - 4x - 6
0 = x² - 2x -3
0 = (x-3) (x+1)
x = 3 or x = -1, but because x>0, x =3
*Example 3: *
3x/x + 1 + 5/ax + 2 = -6x² + 11x + 5/(x + 1) (ax + 2)
In the equation above x≠ -2/a and a is a constant . What is the value of a?
Let's clear the denominators by multiplying both sides by (x+1) (ax + 2):
3x/x+1 × (x +1) (ax +2) + 5/ax + 2 × (x +1) (ax +2) = -6x² + 11x + 5/(x+1) (ax +2) × (x +1) (ax + 2)
3x × (ax +2) + 5 × (x+1) = -6x² + 11x + 5
3x(ax +2) + 5(x + 1) = -6x²+ 11x + 5
3ax² +6x + 5x + 5 = -6x² + 11x + 5
Comparing the coefficients of the x² term on either side, 3a = -6. Therefore, a = -2.
Answer = -2
Algebraic expressions are just a combination of numbers and variables (x² + y)
*1. Combining Like Terms*
When combining like terms, the most important mistake is to avoid putting terms together look like they can go together, but can't. For example, you cannot combine b² + b to make b³, nor can you combine a + ab to make 2ab. To add or subtract variables they have to match.
2(2a² -3a²b² - 4b²) - (a² + 5a²b² - 10b²)
Which of the following is equivalent to the expression above?
2(2a² -3a²b² -4b²) - (a² + 5a²b² -10b²) = 4a² -6a²b² -8b² -a - 5a²b² + 10b²
= 3a² -11a²b² + 2b²
*2. Expansion and Factoring*
2(x-4)(2x + 3)
Which of the following is equivalent to the expression above?
Some people like to factor using FOIL (first, outer, inner, last). Same as distribution. First distribute the 2.
2(x-4) (2x + 3) = (2x -8) (2x + 3)
= 4x² + 6x -16x -24
When it comes to factoring and expansion there are some formulas to know:
1. (a + b)² = a² + 2ab + b²
2. (a - b)² = a² - 2ab + b²
3. a² - b² = (a + b) (a-b)
Which of the following is equivalent to 4x⁴ - 9y²?
The equation above is a different of two squares, a variation of a² - b² formula.
Using the formula a² - b² = (a + b)(a-b), we can see that a = 2x² and b = 3y, Therefore,
4x⁴ - 9y² = *(2x² + 3y) (2x² - 3y)*
16x⁴ - 8x²y² + y⁴
Which of the following is equivalent to the expression above?
Using the formula (a-b)² = a² - 2ab + b² (in reverse), we can see that a = 4x² and b = y², therfore
16x⁴ - 8x²y² + y⁴ = (4x² - y²)²
We have to take it one step further and apply the a² - b² formula to the expression inside the parentheses.
(4x² - y²)² = [(2x + y) (2x -y)]² = (2x + y)²(2x -y)²
*3. Combining Fractions*
When you're adding simply fractions: ½ + ¼
The first step is to find the least common multiple of the denominators.
1/x+2 + 2/x-2
Which of the following is equivalent to the expression above,
The common denominator is just the product of the two denominators: ( x + 2)( x - 2). So now we multiply the top and bottom of each fraction by the factor they don't have:
1/x+2 + 2/x-2 = 1/x+2 (X) x+2/x+2 = x-2/(x+2)(x-2) + 2(x+2)/(x+2)(x-2) = (x-2) + 2(x+2)/(x+2)(x-2)
* = 3x + 2/(x+2)(x-2)*
*4. Flipping (Dividing) Fractions*
What's the difference between ½/3 and 1/2-3
The difference is where the longer fraction line is. The first is 1/2 divided by 3. The second one is 1 divided by 2/3.
½/3 = 1/2 ÷ 3 = ½ ×1/3 = 1/6
½/3 = 1 ÷ 2/3 = 1 x 3/2 = 3/2
The shortcut is to flip the fraction that is in the denominator. So,
a/b-c = ac/b
If the fraction is in the numerator, then the following occurs:
a-b/c = a/bc
If x > 1, which of the following is equivalent to x/1/x-1 + 1/x+1
First combine the two fractions on the bottom with the common denominator (x-1) (x+1)
1/x-1 + 1/x+1 = x+1/(x-1)(x+1) + x-1/(x-1)(x+1) = 2x/(x-1)(x+1)
Next, substitute this back in and flip it.
x/2x/(x-1)(x+1) = x(x-1)(x+1)/2x = *(x-1)(x+1)/2*
*5. Splitting Fractions*
*Example 7:* Which of the following is equivalent to 30 + c/6?
We can split the fraction into two:
30 + c/6 = 30/6 + c/6 = 5 + c/6
This is just the reverse of adding fractions.
Note that while you can split up the numerators of fractions, you cannot do so with denominators
You cannot break up a fraction like 3/x+y any further.
Example 1: The value, V, in dollars, of a home from 2006 to 2015 can be estimated by the equation: V = 240,000 - 5000T, where T is the number of years since 2006.
Part 1: Which of the following best describes the meaning of the number of 240,000 in the equation?
A) The value of the home in 2006.
Part 2: Which of the following best describes the meaning of the number 5,000 in the equation?
B) The yearly decrease in the value of the home.
Many of the scenarios will come in the form: y = mx + b form.
*Example 2:* The maximum height of a plant, h, in inches, can be determine by the equation h = 4x + 6/5, where x is the amount of fertilizer, in grams, used to grow the plant.
Part 1: According to the equation, one more gram of fertilizer would increase the maximum height of a plant by how many inches?
Part 2: To raise the maximum height of plant by exactly one inch, how many more grams of fertilizer should be used in growing the plant?
*Part 1 Solution:* This question is essentially asking for the change in h for every 1 unit increase in x. This is the slope. From the equation, we can see that the slope is 4/5, or 0.8. To make this even clearer, we can put the equation into y = mx + b form by splitting up the fraction: h = 4/5x + 6/5. Note that when we're dealing with changes in x and y, the y-intercept b is irrelevant because it's a constant that's always there.
*Part 2 Solution:* Because this equation is asking for the change in x for every 1 unit increase in h, the reverse of Part 1, we need to rearrange the equation so that we have x in terms of h.
h = 4x + 6/5
5h = 4x + 6
5h - 6 = 4x
x = 5/4h -3/2
Now we can see that x increases by 5/4 or 1.25, when h increases 1. The answer is just the slope of our new equation. A shortcut for this type of question is to take the reciprocal of the slope of the original equation. The reciprocal of 4/5 is 5/4.
T = 65 - 6m
A can of soda is put into a freezer. The temperature, T, of the soda, in degrees Fahrenheit, can be found by using the equation above, where m is the number of minutes the can has been in the freezer. What is the decrease in the temperature of the soda, in degrees Fahrenheit, for every 5 minutes the can is left in the freezer?
The slope of -6 represents the change in the temperature for every 1 minute the can is left in the freezer. So for every 5 minutes the temperature of the soda decreases by 5x6 = 30 degrees Fahrenheit.