Transformations of Quadratics

Directions: • You are going to investigate transforming (“changing”) quadratic functions. We will use the

equation y = x2 as our parent function. • Get out your graphing calculator.

Part 1: Stretching and compressing vertically Graph the following “family” of functions pressing Enter to get to the next equation. A table has been provided to write your coordinate points. Graph your functions on the coordinate plane. 𝑦 = 𝑥$

𝑦 = 2𝑥$

𝑦 = 3𝑥$

In your own words, describe how the parabola changes as the “a’ gets larger.

Write the equation that represents stretching y = x2 by a factor of 6. Now delete the functions you put in your calculator and graph the next “family.” 𝑦 = 𝑥$

𝑦 = '$𝑥$

𝑦 = '(𝑥$

In your own words, describe how the parabola changes if the “a’ is a fraction between 0 and 1.

• Write the equation that represents compressing y = x2 by a factor of 6.

𝑥 𝑦 = 𝑥$ 𝑦 = 2𝑥$ 𝑦 = 3𝑥$

Vertex

𝑥 𝑦 = 𝑥$ 𝑦 =12 𝑥

$ 𝑦 =13 𝑥

$

Vertex

Part 2: Translate Vertically

Now delete the functions you put in your calculator and graph the next “family.”

𝑦 = 𝑥$

𝑦 = 𝑥$ + 3𝑦 = 𝑥$ + 5𝑦 = 𝑥$ − 2

Describe in your own words how to change an equation to translate the parabola vertically: Given the parent function: y = x2, write the equation to translate the parabola vertically up 4 units AND compressed by a factor of 3.

Part 3: Translate Horizontally

Now delete the functions you put in your calculator and graph the next “family.”

𝑦 = 𝑥$

𝑦 = 𝑥 − 3 $

Now try these graphs:

𝑦 = 𝑥$

𝑦 = 𝑥 + 3 $

𝑥 𝑦 = 𝑥$ 𝑦 = 𝑥$ + 3 𝑦 = 𝑥$ + 5 𝑦 = 𝑥$ − 2

Vertex

𝑥 𝑦 = 𝑥$ 𝑦 = 𝑥 − 3 $

1

2

3

4

Vertex

𝑥 𝑦 = 𝑥$ 𝑦 = 𝑥 + 3 $

-4

-3

-2

-1

Vertex

In Function Notation:

𝑓(𝑥) ± 𝑘

How is the new function different from the parent function?

How is the new function different from the parent function?

When you + inside the ( ), the graph moved _______________________. When you – inside the ( ), the graph moved ______________________.

• Write a quadratic function that has shifted left 8 units. • Write a quadratic function that has shifted right 8 units.

Part 4: Reflect over the x-axis

Now delete the functions you put in your calculator and graph the next “family.”

𝑦 =12𝑥

$

𝑦 = −12𝑥

$

Describe in your own words what you notice: Now try these equations in your calculator.

𝑦 = 2𝑥$

𝑦 = −2𝑥$

Describe in your own words what you notice: When a negative is in front, a ___________________________________________ happens. This causes the graph to open ___________.

𝑥 𝑦 =12𝑥

$ 𝑦 = −12𝑥

$

Vertex

𝑥 𝑦 = 2𝑥$ 𝑦 = −2𝑥$

Vertex

In Function Notation:

𝑓(𝑥 ± ℎ)

• Write the equation of a quadratic function that vertically stretches by a factor of 4 and reflect it over the x-axis.

• Describe the translations of the following function: y = 2x2 – 1.

The Whole Shebang J Given the parent function y = x2, write the equation that will shift the parabola to the left 3 units, up 5 units, stretched by a factor of 2 and reflected over the x-axis. Now, can you apply the same thinking using function notation? Describe the transformation applied to the parent function f(x) = x2. 1.

'$ ∙ 𝑓 𝑥 − 2 2. – 𝑓 𝑥 + 5