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Math
Algebra
Study for Math TEST (Unit 1 - Unit 2.1)
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Terms in this set (43)
𝑎^0 = (a to the exponent of 0 equals?)
ANY EQUATION to the EXPONENT of 0 = ?
ex. ( 4(3𝑥𝑦 9 6 ) )^0 = ?
1
a number divided by itself is? ex. a^n / a^n = ? (equation should be seen right side up not left and right)
ALWAYS 1
𝑎^𝑛 / 𝑎^𝑛 = 𝑎^ 𝑛−𝑛 =
a to the exponent of n
divided by
a to the exponent of n = is equivalent to a to the exponent of n-n
this equation's answer is a = ?
a = 0
5^0 (five to the exponent of 0) = ?
1
−7(𝑥^3 )^0 =
-7(x to exponent of 3)to the exponent of 0 = ?
hint do the inside parenthesis first.
-7( 1 ) = when simplified
answer: -7
x^3 times (multiplied by) x^4 = ?
how many x's is that in total?
7 x's or x * x * x * x * x
x
x
or x * x * x * x * x
x
x = x^7 (cause you add them)
ex. 2^3 * 2^4 = 2^ ?
(can also be done on the CALCULATOR)
2^7 (because you add them)
y^9 / y^5 = ?
9-5 = 4 so y^4
when dividing exponents, subtract.
the exponents cancel each other out until there are exponents left.
(a^m)^n
(a to the exponent of m in parenthesis) raised to the exponent of n =
a ^ m * n
a raised to the exponent of
m multiplied by n
h^-2
h to the exponent of -2 =
1
/
h ^ 2
1 over or divided by h to the exponent of POSITIVE 2
7 ^ 1/3 = ?
(7 to the exponent of 1/3) = ?
3√7
x ^ 3/7 = ?
x to the exponent of 3/7 = ?
7 √ x^3
7 on the outside of the square root, x to the exponent of 3 on the inside of the square root
If the input value is 2
what does f(x) = x^2 + 3x − 4 become?
f (2) = 2^2 + 3(2) − 4
which equals 6 in all if you calculate it
f (a + h) = (a + h)^2 + 3(a + h) − 4
what do you do to solve this?
distributive property!
a^2 + 2ah + h^2 + 3a + 3h −4
^3 √ 4 ^3 = ?
radical 3 square root 4 to the exponent of 3
you CAN do this on the CALCULATOR!
how do you do it on the calc?
click MATH go to the number 5
Simplify
√300
remember to FACTOR
when factoring it becomes
√100 ∙ √3
which can be factored again into
10√3
Simplify the radical expression √162 a^5 * b^4
(all variables are INSIDE the square root btw its all one square root.)
First factor the radicand into a perfect square factor and another factor, and write the radical expression as a product of radical expressions.
(Notice that a must be positive in order for the original expression to be defined.)
√81 a^4 b^4 * √2a
when i do the square root the variables are ALL in it
simplified again
9a^2*b^2 √2a
(9 a to the exponent of 2 and b to the exponent of 2 OUTSIDE of the square root)
you can simplify it again but teacher did not give answer to the problem so.. lame
As long as a is positive, we know that √a ∙ √a = ?
√a^2
square root of a to the exponent of 2
which IN ALL equals just plain a
a ^1/2 * a^1/2 = ?
just a ..
= a
(radical of 3s so the tiny 3 not 3 square root.)
3 √ a
3 √ a
a * 3 √ a = ?
radical of 3 √ a^3
which equals plain a.
=a
radical of n√a = is also equivalent to
a^1/n
a to the exponent of 1/n
3 radical √x = ?
x can be any real number
x ^ 1/3
x to the exponent of 1/3
interval notation when combining more than 1 set of numbers is used with a __
U
f(x) = x+1 / 2-x
x + 1
/
2 - x
simplify it
first 2 - x = ?
what do we do to solve this?
make 2 - x = 0
make it equal to 0 to solve for x.
you should get
- x = - 2
which is x = 2 in the end.
in interval notion this would be
(-infinity, 2) U (2, infinity)
when multiplying by a negative number you must ____ the inequality signs
−x ≥ −7
REVERSE
x ≤ 7
what do you do to find domain from graphs?
hint: go from left to right (horizontally for domain)
when you see the end point stops on an x value (on the left or where ever the end point is)
that is your domain
and then if it stops at a different number on the right side that will be your right interval
these can also be infinities
ex. [-5, infinity)
how do you find range on a graph?
up and down!
look at what number your lines stop at (end points)
example if you see the line is all the way up and coming down on the left side with no end point, your range is
from the left (infinity, ___)
absolute value function looks like?
cubic function looks like?
reciprocal function looks like?
reciprocal squared function looks like?
same as a reciprocal function but on the same top axis
so like from quadrant 1 to quadrant 2 (left to right)
not opposite like reciprocal function.
root function looks like?
cube root function looks like?
maximums and minimums on a graph
how do you tell?
the max and the min are the highest point and the lowest point on the graphs
increasing and decreasing graphs
from the graph you can see that it is increasing from
(1,3)
Notice that when a graph is bending upward over an interval,
we say that it is concave up
When a graph is bending downward over an interval,
we say that it is concave down.
f^-1(x) is called an ____
inverse function
when dealing with inverse functions
the output becomes the input and the input becomes the output, so theyre switched.
solve for the inverse function of
f(x)= 2/x-3 + 4
2
/ + 4
x-3
what should you make
f(x) into?
you should make it
y=
so
the equation becomes
y= 2/x-3 +4
subtract 4 from both sides
then multiply both sides by x-3 and divide by y-4
add 3 to both sides
x= 2/y-4 + 3
now your answer can be put as
f^-1 (y) = 2/y-4 + 3
exponential growth formula
a is?
b is?
f(x)=ab^x
a is the starting or initial value
and b is the growth factor
when finding the percents
increasing is + 1
and decreasing is - 1
given the equation for exponential functions what is the y int of
f(x) = 4(0.95^x)
and
f(x)=7(1.02^x)
(0,4)
and
(0,7)
the first letters of the equations are the y
intercepts :)
when looking at a table that has
f(0) = 160
f(1) = 136
f(2) = 115.6
what is the starting value?
and how do you calculate what it is constant by?
in the end what is the equation?
the initial value is what you see is =0
so the starting value is 160.
you DIVIDE to see what the constant factor is.
in this example the constant factor is 0.85
in the end the formula of the function is
f(x)=160(0.85)^x
how do you find exponential equation with 2 points?
what is the formula?
the formula is:
f(d) ab^d
/ =
f(c) ab^c
f(d)/f(c)= ab^d/ab^c
example:
(x,y)
(2,61) (5,26)
26/61 = ab^5/ab^2
solve for b by canceling the a's
b^5-2 = b^3
26/61 = b^3
take the cube root of both sides.
cube rooting is the same as raising to the power of 1/3
so
(26/61)^1/3 = (b^3)^1/3
eventually you get b=3.5
now that we have 3.5
put it into the formula of
f(x)= ab^x
f(x)= a(3.5)^x
use the first set of numbers (2,61)
61=a(3.5)^2
divide both sides by 3.5
and you get
a= 61/3.5 = 5
the equation ends up being :
f(x)=5(3.5)^x
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