Chapter 2, Section 1: Using Transformations to graph quadratic functions.
A quadratic function can be written in the form 𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘. How this equation is different from
absolute value functions, 𝑓(𝑥) = 𝑎 |𝑥 − ℎ| + 𝑘, is the shape of the graph. Just like with absolute value functions,
a, h, and k represent numbers. Some sample equations may look like this
Absolute Value Function
Quadratic Function
𝑓(𝑥) = |𝑥 + 2|
𝑓(𝑥) = (𝑥 + 2)2
𝑓(𝑥) = |𝑥| − 3
𝑓(𝑥) = (𝑥)2 − 3
𝑓(𝑥) = |𝑥 − 4| + 1
𝑓(𝑥) = (𝑥 − 4)2 + 1
𝑓(𝑥) = 2|𝑥|
𝑓(𝑥) = 2(𝑥)2
The shape of the absolute value function was a “V”. The shape of any quadratic function is a parabola. The basic
shape can be found by graphing 𝑓(𝑥) = (𝑥)2 . The point
where the graph has its minimum (or maximum) value, is
called the vertex. This is just like the point of the “V” in
absolute value graphs.
How to graph with the calculator. To graph 𝑓(𝑥) = (𝑥 + 5)2 − 3, open a
new document and add a graph page.
To enter the equation, use the left parenthesis button and the x2 button.
Your graph and equation should look like the screen below.
To trace the graph, press MENU, select 5: Trace, the select 1: Graph
Trace. Using the left and right arrows will let you trace around on the
graph. This shows a vertex at (-5,-3) as a minimum.
Activity 1. Finding the vertex. For each of the following equations, write what you think the coordinate of the
vertex will be. Then, type the equations into the graphing calculator to check your answer.
1. 𝑓(𝑥) = (𝑥 + 2)2 + 1
vertex = (
,
)
2. 𝑓(𝑥) = (𝑥 − 3)2
vertex = (
,
)
3. 𝑓(𝑥) = (𝑥 + 4)2 − 2
vertex = (
,
)
4. 𝑓(𝑥) = (𝑥 + 3) − 5
vertex = (
,
)
5. 𝑓(𝑥) = 𝑥 2 + 3
vertex = (
,
)
2
6. Compare your results with your initial guess. Explain, with complete sentences, how to determine the vertex
of a parabola, without a calculator, if you are given the equation.
Activity 2. The effect of a. What effect does the number in front of x2 have on the graph? Enter the following
equations into the graphing calculator one at a time and draw a sketch on the grid provided.
7. 𝑓(𝑥) = (𝑥)2
8. 𝑓(𝑥) = 3(𝑥)2
9. 𝑓(𝑥) = 10(𝑥)2
10. 𝑓(𝑥) = 0.5(𝑥)2
11. What do all the graphs have in common?
12. What effect does having a value of a > 1 have
on the graph?
13. What effect does having a value of a < 1 have on the graph?
Write the coordinates of the vertex for each of the following equations.
1. 𝑓(𝑥) = 𝑥 2 − 7
vertex = (
,
)
2. 𝑓(𝑥) = (𝑥 − 9)2
vertex = (
,
)
3. 𝑓(𝑥) = (𝑥 + 7)2
vertex = (
,
)
4. 𝑓(𝑥) = (𝑥 − 5)2 + 8
vertex = (
,
)
5. 𝑓(𝑥) = 3𝑥 2 − 2
vertex = (
,
)
6. 𝑓(𝑥) = −(𝑥 + 4)2 − 6
vertex = (
,
)
7. 𝑓(𝑥) = 2(𝑥 − 7)2
vertex = (
,
)
8. 𝑓(𝑥) = (𝑥 − 4)2 − 3
vertex = (
,
)
9. 𝑓(𝑥) = −(𝑥 + 4)2 − 2
vertex = (
,
)
10. 𝑓(𝑥) = 2 (𝑥 + 4)2 − 2
vertex = (
,
)
11. 𝑓(𝑥) = 𝑥 2 + 5
vertex = (
,
)
12. 𝑓(𝑥) = 2(𝑥 − 3)2
vertex = (
,
)
13. 𝑓(𝑥) = −(𝑥 + 1)2 − 4
vertex = (
,
)
14. 𝑓(𝑥) = (𝑥 − 4)2 + 1
vertex = (
,
)
15. 𝑓(𝑥) = −2(𝑥 − 8)2 + 3
vertex = (
,
)
16. 𝑓(𝑥) = −𝑥 2 − 8
vertex = (
,
)
17. 𝑓(𝑥) = (𝑥 + 3)2
vertex = (
,
)
18. 𝑓(𝑥) = (𝑥 − 1)2 − 1
vertex = (
,
)
19. 𝑓(𝑥) = 4(𝑥 + 3)2 − 2
vertex = (
,
)
vertex = (
,
)
1
1
20. 𝑓(𝑥) = − 2 (𝑥 − 7)2 + 3