7.6 Vertex Form of a Quadratic Function
Vertex-graphing form :
y = a(x – h)2 + k
where a, h, k are real numbers and a ≠ 0.
Vertex: the vertex is the point on the graph where the graph changes direction (from going down to up, or up to down)
The vertex is labeled as (h,k) where h and k are real numbers
Warm Up – Access the online graphing calculator to answer the following questions
1. Graph the following function 𝑦 = 𝑥 2 .
Change the graph by changing the coefficient of x2. Try 2 positive and 2 negative values. How do the parabolas
change as you change this coefficient?
Coefficient:
a. ______
b. ______
c. ______
d. ______
Resulting change:
___________________________
___________________________
___________________________
___________________________
2. For each function you graphed in part 1, determine the coordinates of the vertex and the equation of the axis of
symmetry.
Vertex:
a. ______
b. ______
c. ______
d. ______
Axis of symmetry:
___________________________
___________________________
___________________________
___________________________
3. Graph the following function 𝑦 = 𝑥 2 + 1
.
Change the graph by changing the constant. Try 2 positive and 2 negative values. How do the parabolas change as
you change this constant? How do the coordinates of the vertex and the equation of the axis of symmetry change?
Constant:
a. ______
b. ______
c. ______
d. ______
Resulting change:
___________________________
___________________________
___________________________
___________________________
Vertex:
(_______)
(_______)
(_______)
(_______)
Axis of symmetry:
_______________
_______________
_______________
_______________
4. Graph the following function 𝑦 = (𝑥 − 1)2 .
Change the graph by changing the constant. Try 2 positive and 2 negative values. How do the parabolas change as
you change this constant? How do the coordinates of the vertex and the equation of the axis of symmetry change?
Constant:
a. ______
b. ______
c. ______
d. ______
Resulting change:
___________________________
___________________________
___________________________
___________________________
Vertex:
(_______)
(_______)
(_______)
(_______)
Axis of symmetry:
_______________
_______________
_______________
_______________
5. The equation we used was expressed in vertex form: 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘.
How do the values of a, h, and k determine the characteristics of the
parabola?
a
h
k
Sketching the graph of a quadratic function given in vertex form
Ex. Sketch the graph of the following function:
𝑓(𝑥) = 2(𝑥 − 3)2 − 4
State the domain and range of the function.
Determining the equation of a parabola using its graph
Ex. Liam measured the length of the shadow that was cast by a meter stick at
10 a.m. and at noon near his home in Saskatoon and found the shadow was
85.3m and 47.5m respectively. After discussing with his class, they decide the
shadow was the shortest at noon. Try predict the equation using Liam’s
information. Draw a graph to help you.
Reasoning about the number of zeros that a quadratic function will have
Ex. Predict whether a quadratic function will have zero, one, or two zeros if the function is expressed in vertex form. List
them if they exist.
a. 𝑓(𝑥) = 2(𝑥 − 2)2 − 5
b. 𝑓(𝑥) = 𝑥 2
c. 𝑓(𝑥) = 2(𝑥 + 3)2 + 4
Solving a problem that can be modeled by a quadratic function
Ex. A soccer ball is kicked from the ground. After 2 s, the ball reaches its
maximum height of 20 m. it lands on the ground at 4 s.
a. Determine the quadratic function that models the height of the kick
b. Determine any restrictions that must be placed on the domain and range of
the function.
c. What was the height of the ball at 1 s? When was the ball at the same height
on the way down.