Transformations (Stretch, Compress, Reflect)

Transformations (Stretch, Compress, Reflect)

We will describe the effects of changes to the parent function x²

Essential Question

How can you describe the graph of f(x) compared to the parent graph of f(x), if a is a number greater than zero?

Let's step back....

Lets make sure we can describe some characteristics of a quadratic function including the domain and range...

Multiple Choice

What is the maximum/minimum of a parabola called?

• vertex

• point

• top

• bottom

Multiple Choice

Does the equation open up or down?

Y = -3x² +7x - 2

• up

• down

• Neither; it opens to the right.

• Neither; it opens to the left.

Multiple Choice

What is the axis of symmetry?

• the slope of the graph

• the dividing line for a parabola

• a way to spin my pencil

• the x-axis

Multiple Choice

What is the range of this function?

• All Real Numbers

• $0\le x\le4$

• $y\ge-2$

• $y\ge2$

Multiple Choice

Does this graph in the back have maximum or minimum value?

• maximum

• minimum

• neither

• both

Multiple Choice

What is the graph of quadratic function?

• a. circle

• b. square

• c. parabola

• d. ellipse

Multiple Choice

What is the vertex of y=x²+4x+3?

• (2,1)

• (-2,1)

• (0,0)

• (-2,-1)

Multiple Choice

What is the domain and range of the following function:
y = 3x² -6x +5

• Domain: All Real Numbers
  Range:  $y\le-2$  

• Domain: All Real Numbers
  Range:  $y\ge-2$  

• Domain: All Real Numbers
  Range:  $y\ge2$  

• Domain: All Real Numbers
  Range:  $y\le2$  

Before we start...

If you don't take out something to take some notes with...

Stretch, Compress, Reflect

We will look at what make a quadratic funtion stretch or compress.

Vertex Form

*Remember we get the vertex from

 $x=-\frac{b}{2a}$  , in standard form  $ax^2+bx+c$  .

In case you forgot...

Use the Standard form of a quadratic equation  $ax^2+bx+c$  .  After you calculate the x-coordinate, insert back into f(x).  So  $f\left(-1\right)=\ 2\left(-1\right)^2+4\left(-1\right)+5$  ,  $f\left(-1\right)=3$  

So the vertex (h,k) is (-1,3)

We will focus on "a"

The coefficient "a" will determine both reflections across the x-axis (open up or down) and a stretch or compression.

We will focus on "a"

The parent function is  $f\left(x\right)=x^2$  .  Every function can be produced by applying changes to this  main function.  
A Reflection across the x-acis will occur if  $-f\left(x\right)=-\left(x^2\right)=-x^2$  
*If a is negative then reflect (down)

We will focus on "a"

A horizontal stretch or compression occurs when a is a rational coefficient  $\frac{1}{b}$  , the pay close attention to b.  

 $\left|b\right|>1$  widens the graph.  
 $0<\left|b\right|<1$  squeezes the graph.

We will focus on "a"

A vertical stretch or compression occurs when..

 $f\left(x\right)=x^2$  
 $af\left(x\right)=ax^2$  
 $\left|a\right|>1$ makes the graph "taller" . (Pulls away from the x-axis)
 $0<\left|a\right|<1$  smashes the graph.  Think flatten downward.

Lets take it slowly

Lets see if we can identify a reflection first then we can move to stretches and compressions.

Multiple Choice

Which transformation on

f(x) = x

is g(x) = -f(x)

• Reflection across the y-axis

• The slope will be less steep

• The graph will be wider

• Reflection across the x-axis.

Multiple Choice

How did we transform from f(x) =x²

to g(x) = -3x²

• reflection in x-axis and vertical shift down

• reflection x-axis and vertical stretch

• horizontal stretch

• reflection x-axis and vertical compression

Multiple Choice

Does the graph of this equation open up or down? 
f(x) = -(x + 3)² - 5

• up
• down

Multiple Choice

Vertical compression is af(x) when a is...

• a fraction/decimal and the graph flattens

• a number greater than 1 and the graph becomes steeper

• any type of number

• any number less than 1

Multiple Choice

Vertical stretch is af(x) when...

• a is greater than 0
• a is less than 0
• a is greater than 1
• a is a fraction/decimal

Multiple Choice

af(x) is a...

• Vertical translation
• Horizontal stretch/compression
• Horizontal translation
• Vertical stretch/compression

Multiple Choice

Compared to the parent function, f(x) = x², which of the following is the equation of the function after a vertical shrink by a factor of 1/3?

• A
• B
• C
• D

Multiple Choice

Compared to the parent function, f(x) = x², which of the following is the equation of the function after a vertical stretch by a factor of 3?

• A
• B
• C
• D

Multiple Choice

How does -1/5 affect the parent function?

g(x) = -1/5(x - 1)² + 7

• reflection, vertical compression

• vertical compression, horizontal shift left

• reflection, horizontal shift right

• no changes were made to y = x²

Open Ended

How can you describe the graph of f(x) compared to the graph of f(x), if a is a positive number. For example: af(x) = 2x²