Lesson 2 Transformers: More Than Meets the y’s Solidify Understanding

Jump Start

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$f (x) = x^{2} - 3$
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$f (x) = 3 x^{2}$
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$f (x) = {(x - 3)}^{2}$
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$f (x) = \frac{1}{3} x^{2}$

Learning Focus

Write equations for functions that are transformations of $f (x) = x^{2}$ .

Find efficient methods for graphing transformations of $f (x) = x^{2}$ .

What happens to the graph of $f (x) = x^{2}$ when more than one transformation is applied?

Open Up the Math: Launch, Explore, Discuss

In the previous lesson, you learned about transformations of the graph of $f (x) = x^{2}$ . In this lesson, you will be writing equations and graphing functions that have more than one transformation. It will be a lot easier if you know a few points on $f (x) = x^{2}$ that we can use for comparisons. We’ll call them anchor points.

1.

Identify each anchor point shown on the graph of $f (x) = x^{2} .$

Anchor points:

Vertex $(\underset{―}{}, \underset{―}{})$
$(1, \underset{―}{})$ and $(- 1, \underset{―}{})$
$(2, \underset{―}{})$ and $(- 2, \underset{―}{})$
$(3, \underset{―}{})$ and $(- 3, \underset{―}{})$

Line of symmetry: $x = \underset{―}{}$

Write the equation for each problem below. Use a second representation to check your equation.

2.

The area of a square with side length $x$ , where the side length is decreased by $3$ , the area is multiplied by $2$ , and then $4$ square units are added to the area.

3.

4.

$x$	$f (x)$
$- 4$	$7$
$- 3$	$2$
$- 2$	$- 1$
$- 1$	$- 2$
$0$	$- 1$
$1$	$2$
$2$	$7$
$3$	$14$
$4$	$23$

5.

Graph each equation without using technology. Be sure to have the exact vertex and at least two correct points on either side of the line of symmetry.

6.

$f (x) = - x^{2} + 3$

7.

$g (x) = {(x + 2)}^{2} - 5$

8.

$h (x) = 3 {(x - 1)}^{2} + 2$

9.

Given: $f (x) = a {(x - h)}^{2} + k$

a.

What point is the vertex of the parabola?

b.

What is the equation of the line of symmetry?

c.

How can you tell if the parabola opens up or down?

d.

How do you identify the vertical stretch?

10.

Does it matter in which order the transformations are done? Explain why or why not.

Ready for More?

Think about applying the transformations to the parent function $f (x) = 2^{x}$ .

1.

What point makes sense to use as an anchor point on this function?

2.

What do you think is the equation of the function with a horizontal shift left $3$ ?

3.

How does the horizontal shift on $y = 2^{x}$ work like the horizontal shift on $y = x^{2}$ ?

Takeaways

Vertex form of a quadratic equation:

Vertex:
Line of symmetry:
Vertical stretch:

Opens upward:
Opens downward:

Quick-graph method for graphing quadratics:

a. $y = {(x - 4)}^{2} - 3$

c. $y = 2 {(x + 1)}^{2} - 5$

b. $y = - {(x + 2)}^{2} + 5$

Vocabulary

vertex form
Bold terms are new in this lesson.

Lesson Summary

$x$	$y$
$- 4$	$9$
$- 3$	$2$
$- 2$	$- 3$
$- 1$	$- 6$
$0$	$- 7$
$1$	$- 6$
$2$	$- 3$

Vertex: $\underset{―}{}$
Line of symmetry: $\underset{―}{}$
$x$ -int(s): $\underset{―}{}$
Minimum or maximum?