Notes 4.16 - Vertex Form
Name________________________ Per____
Using what you know about transformations, match each parabola to its equation in vertex form and factored
form. Also write the vertex and roots (x-intercepts) of the parabola.
Bank of Equations
y =( x + 4) − 4
y=
−( x − 4) + 4
y =( x − 4) 2 − 4
y=
−( x + 4) 2 + 4
( x − 2)( x − 6)
y=
( x + 2)( x + 6)
y=
y=
−( x + 2)( x + 6)
y=
−( x − 2)( x − 6)
2
2
Graph 1
Graph 2
Graph 3
Graph 4
Vertex Form
Vertex Form
Vertex Form
Vertex Form
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Vertex
Vertex
Vertex
Vertex
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Factored Form
Factored Form
Factored Form
Factored Form
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Roots
Roots
Roots
Roots
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1. How does vertex form relate to the graph and the parabola’s vertex?
2. How does factored form relate to the graph and the parabola’s roots?
3. Are any of these parabolas stretched vertically or horizontally? How do you know?
Vertex Form of a Quadratic Equation: y = a ( x − h) + k
2
Graph each of the following parabolas using transformations.
4. f ( x) = x
*this is the parent graph
5. g ( x) =( x − 2) + 3
−( x + 1)
6. h( x) =
7. j ( x) = 2( x − 3) − 6
2
2
2
2
8.
9.
Without graphing, state the vertex for each of the following quadratic equations.
a.
𝑦 = (𝑥 − 5)2 + 3
b.
𝑦 = 𝑥 2 − 2.5
c.
𝑦 = (𝑥 + 4)2
Write a quadratic equation whose graph will have the given vertex.
d.
(1.9, −4)
e.
(0, 100)
f.
�−2,
3
�
2
Write each of the following vertex form equations in standard form.
10. y = ( x − 1) + 8
2
−( x + 3) − 2
11. y =
2
−3( x − 4)
12. y =
2
Write each of the following standard form equations in vertex form.
13. y = x − 4 x + 7
2
14. y = 2 x + 12 x − 3
2
− x + 12 x − 19
15. y =
2
16. Caitlin has 60 feet of material that can be used to make a fence. Using this material, she wants to create a rectangular
pen for her dogs to play in. What dimensions will maximize the area of the pen?
a.
Let 𝑤 be the width of the rectangular pen in feet. Write an expression that represents the length when the width is
𝑤 feet.
b.
Define a function that describes the area, 𝐴, in terms of the width, 𝑤.
c.
Rewrite 𝐴(𝑤) in vertex form.
d.
What are the coordinates of the vertex? Interpret the vertex in terms of the problem.
e.
What dimensions maximize the area of the pen? Do you think this is a surprising answer?
Lesson Summary
When graphing a quadratic equation in vertex form, 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘, (ℎ, 𝑘) are the coordinates of the vertex.