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Math - Graphing & Solving Quadratics
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Terms in this set (37)
What is the standard form of a quadratic equation?
y = ax² + bx + c
What is a parabola?
u-shaped curve created by a quadratic function
What is a minimum?
vertex is at its lowest point
What is a vertex?
turning point of the parabola
What is a maximum?
vertex is at its highest point
If "a" is positive, the parabola opens ______. The vertex point is called a ________.
up; minimum
like a smile :)
If "a" is negative, the parabola opens ______. The vertex point is called a ________.
down; maximum
like a frown :(
What is an axis of symmetry? What is the equation?
the vertical line that divides the parabola into two symmetric (equal) parts
How do you graph a quadratic equation?
1. Find the axis of symmetry.
2. Find the vertex by plugging in the axis of symmetry (x) into the original equation & solving for "y." Write vertex as coordinates.
3. Fill in a table of values to figure out where to plot points. Pick a couple of numbers above or below the "x" value of your vertex and plug it into the original equation to solve for "y."
4. Reflect these points across the axis of symmetry and draw your parabola.
y = x² + 8x + 15
What is the axis of symmetry and vertex? Does the parabola open up or down?
AOS: x = -4
V: (-4, -1)
opens up
y = -x² + 10x - 23
What is the axis of symmetry and vertex? Does the parabola open up or down?
AOS: x = 5
V: (5, 2)
opens down
y = -2x² + 12x - 15
What is the axis of symmetry, vertex, domain, and range? Does the parabola open up or down?
AOS: x = 3
V: (3, 3)
D: all real numbers
R: y ≤ 3
opens down
y = 3x² - 5
What is the axis of symmetry, vertex, domain, and range? Does the parabola open up or down?
AOS: x = 0
V: (0, -5)
D: all real numbers
R: y ≥ -5
opens up
y = x² - 2x
What is the axis of symmetry, vertex, domain, and range? Does the parabola open up or down?
AOS: x = 1
V: (1, -1)
D: all real numbers
R: y ≥ -1
opens up
What is the domain and range for quadratic equations?
DOMAIN
- all real numbers
RANGE
- if the parabola opens up, y ≥ [y value in vertex]
- if the parabola opens down, y ≤ [y value in vertex]
When b = 0 in your equation....
- axis of symmetry is x = 0
- y value in vertex is equal to "c"
EX: y = x²
- 3
- AOS: x = 0
- V: (0,
-3
)
How do you solve quadratics by the square root method?
1. Isolate x²
2. Take the square root of both sides. Remember, every square root has 2 answers - positive & negative.
3. Write answer in this form: x = ± ? (means both positive and negative)
What does it mean for a quadratic equation to have no solution, 1 solution, or 2 solutions?
- no solution = parabola does not intersect x-axis
- 1 solution = vertex is on x-axis
- 2 solutions = parabola intersects twice with x-axis
Solve this quadratic equation by the sq. root method:
x² + 25 = 0
no real solution (you can't take the root of a negative)
Solve this quadratic equation by the sq. root method:
x² - 49 = 0
x = ± 7
Solve this quadratic equation by the sq. root method:
-2x² = -98
x = ± 7
Solve this quadratic equation by the sq. root method:
3/4x² = 12
x = ± 4
Solve this quadratic equation by the sq. root method:
9x² - 16 = 0
x = ± 4/3
Solve this quadratic equation by the sq. root method:
4x² - 1 = 24
x = ± 5/2
Solve this quadratic equation by the sq. root method:
49x² - 3 = 13
x = ± 4/7
Solve this quadratic equation by the sq. root method:
5x² - 15 = 45
x = ± 3.46
OR
x = ± 2√3
How do you solve a quadratic equation through factoring?
1. Write the equation in standard form.
2. Set the quadratic equation equal to 0. (ax² + bx + c = 0)
3. Factor the left side.
4. Set each factor equal to 0 and solve for x.
5. Write your answer using curly braces {}. List the numbers from least to greatest.
Solve this quadratic equation by the factoring method:
x² + 4x + 3 = 0
{-3, -1}
Solve this quadratic equation by the factoring method:
x² + 1x - 2 = 0
{-2, 1}
Solve this quadratic equation by the factoring method:
x² + 10x + 25 = 0
{-5}
Solve this quadratic equation by the factoring method:
3x² + 15x = 0
{-5, 0}
Solve this quadratic equation by the factoring method:
4x² = 28x
{0, 7}
Solve this quadratic equation by the factoring method:
11x² = x² + 8x
{0, 4/5}
Solve this quadratic equation by the factoring method:
16x² = 9
{-3/4, 3/4}
What are the x-intercepts of the graphs?
solution to the quadratic equation
The x-coordinate of the vertex is the same as...
the axis of symmetry (since it goes through the vertex)
Roots, x-intercepts, and solutions mean (the same/different) things.
the same
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