Transformation of Parent Functions

Transformations in the Coordinate Plane

Parent Functions edition

Multiple Choice

What is the name of the parent function and equation for the graph?

• Rational - y = 1/x
• Absolute Value - y = |x|
• Linear - y = x
• Quadratic - y = x²

Multiple Choice

Name the parent function. 

• constant
• square root
• linear
• absolute value

Multiple Choice

Name the parent function.

• quadratic
• cubic
• square root
• logarithmic

Multiple Choice

Which parent function matches the graph?

• y = |x|
• y = x
• y = x²
• y = 2^x

When you make a change to the x inside the parentheses or absolute value

Move graph left or right, SLIDING opposite direction of the sign. LEFT FOOT UP, RIGHT FOOT SLIDE! So, if the negative "foot" is up, slide to the positive right. Left "foot" up, right foot slide. Can't let this one slide...

Quadratics Parent function  $y=x^2$  

Horizontal translations

•  $y=\left(x+2\right)^2$  Notice the change to the original parent function is INSIDE parentheses and we moved 2 left, since the sign was positive. The blue line moved 4 to the right because of the -4 inside the parentheses.

Multiple Choice

The blue graphs is the parent function f(x)=|x|, the red must be

• g(x)=|x+2|

• g(x)=|x|+2

• g(x)=|x-2|

• g(x)=|x|-2

Multiple Choice

Tell how the graph transformed.
y = (x - 3)²

• Translated right 3
• Translated left 3
• Translated up 3
• Translated down 9

Up and down movement of a function is called a vertical translation.

• When something is added OUTSIDE parentheses or absolute value, it shows vertical movement.

• The black graph moved up five units from the parent function, so we know the equation for this would be:

•  $y=x^2+5$  

Vertices

• The parent function  $x^2$  has a vertex at the origin, or (0, 0)

•  $y=x^2+5$  The movement upward changes the vertex of the new function to (0,5)

Multiple Choice

If the blue is f(x)=x^2, then the red must be

• g(x)=x²-5
• g(x)=x²+5
• g(x)=(x-5)²
• g(x)=(x+5)²

Multiple Choice

(0,0) is vertex of blue parent f(x)=x² _{What is the vertex of the new, red function?}

• (5, 0)

• (0, -5)

• (0, 0)

• (5, 5)

Multiple Choice

• Left 2, Down 2, Vertical Stretch
• Right 2; Up 1; 
• Vertical Stretch; Right 2; Up 1
• Left 2; Down 2;

Multiple Choice

• f(x)=x² + 2
• f(x) = (x-2)² + 1
• f(x)= (x+2)² - 1
• f(x)= (x+2)² + 1

 $y\ =\ \left(x-2\right)^2+1$  

• Inside parentheses change - opposite sign, so move to the right two

• Addition outside parentheses, move up 1

• Results in a vertex of (2, 1) Are you starting to see the pattern?

Rewind your mind 🔙 with the linear parent function & changes in slope.

• Slope y=mx+b, slope is m

• y=2x steeper slope

• y=x linear parent function

• y=1/2x wider slope

Rewind your mind 🔙 with the absolute value parent function & dilations

• Dilations are also known as a stretch or a shrink and change the size of the graph, where translations just slid the graph's position and size stayed same.

• y=|2x| steeper, "thinner," stretched up, vertical stretch

• y=|x| absolute value parent function

• y=|1/2x| wider, "smooshed down," shrunk, vertical compression

Dilations work the same in quadratics.

• Dilations are also known as a stretch or a shrink.

• y=4x^2 "skinnier," more vertically stretched dilation

• y=x^2 quadratic or square parent function

• y=(1/4x)^2   wider, "shrunken" version of the parent. Compression

• y= - x^2 results in a reflection across the x axis.  

Multiple Choice

f(x) = 2x

• Vertical stretch

• Vertical compression

• Vertical translation up

• Reflection over the x-axis

• No Transformation

Multiple Choice

Compare the function y = 0.3x² to the parent function y = x²

• Wider
• Narrower

Multiple Choice

Compare the function y = 5x² to the parent function y = x²

• Wider
• Narrower

Reflections

• a negative version of the function

• y = x^2 quadratic parent function

• y = -x^2 "flips"the parent upside down like a reflection in a pool

Absolute Value Reflection

• y = |x| parent function

• y = -|x| reflection of parent

Multiple Choice

What transformation is represented by

$$-f\left(x\right)$$  ?

• X axis reflection

• Y axis reflection

• Vertical Stretch

• Vertical Shrink

Calculator Graphing

• Hit y= and enter your function

• Then hit graph.

• Each graph is color coded so you can easily see what is what.

• Oh yeah... ABSOLUTE VALUE is 2nd 0. You'll see Abs is first thing in catalog.

Multiple Select

Which of the following create congruent images? Select all that apply.

• Translations

• Reflections

• Rotations

• Dilation

Multiple Select

What transformations have been applied to the function

$$f\left(x\right)=-\left(x\right)^2+3$$  ? Select all that apply.

• Reflection over the x axis

• Reflection over the y axis

• Up 3

• Right 3

RED CHILD: $-\left(x^2\right)+3$         BLUE parent:  $x^2$      

• The negative sign in front of the x^2 vertically flips the parent function (BLUE) over the x axis. 

• The +3 outside the function moves it up three units from the origin.

• Vertex of child? (0, 3)

Multiple Choice

Which of the following does NOT contain a vertical translation DOWN of y = x²?

• y = 2(x - 5)² - 3
• y = -5(x + 3)² - 2
• y = (3x - 2)² + 5
• y = -3(x - 2)² - 5

Multiple Choice

If the function f(x) = x² is changed so that a new function is created, g(x) = 5x², how does g(x) compare to f(x)?

• The graph of g(x) is wider than the graph of f(x)

• The graph of g(x) is narrower than the graph of f(x)

• The graph of g(x) is shifted 5 units up above f(x)

• The graph of g(x) is shifted 5 units down below f(x)

Multiple Choice

If red represents y = x², which equation represents blue?

• y = 2x²
• y = -2x²
• y = -x²
• y = (x - 1)²

Multiple Choice

In the equation f(x)=5(x+3)²-10 what does the 5 do?

• Vertical stretch by 5
• Horizontal compress by 5
• Reflect
• Move up 5

3 transformations: f(x)=5(x+3) - 10

• The 5 in front caused a vertical stretch of the parent graph. Narrower than the blue parent  $x^2$  

• The addition in the parentheses moves the graph to the left (opposite the sign).

• -10 outside the function moves graph down 10

• The vertex of the parent function is (0,0) Vertex of the child? (-3, -10) Can you see those numbers in the equation?

Multiple Select

Given the parent function

$$f\left(x\right)=\left|x\right|$$  describe the transformations that have occurred in  $$g\left(x\right)=-\left|x\right|+3$$  

• Vertical Reflection

• Horizontal Reflection

• Shift Right

• Shift Up

• Shift Down

Multiple Choice

Which function has been shift right 2 and up 3?

• $$f\left(x\right)=\left|x-2\right|+3$$

• $$f\left(x\right)=\left|x+2\right|-3$$

• $$f\left(x\right)=\left|x-2\right|-3$$

• $$f\left(x\right)=\left|x+2\right|+3$$

Multiple Choice

Which function has been vertically compressed by 1/2 and translated up 1?

• $$f\left(x\right)=\frac{1}{2}\left|x\right|+1$$

• $$f\left(x\right)=\frac{1}{2}\left|x-1\right|$$

• $$f\left(x\right)=\left|\frac{1}{2}x\right|+1$$

• $$f\left(x\right)=\left|2x\right|+1$$

Multiple Choice

What is the equation of the absolute value function?

• f(x)= |x+4|
• f(x)= |x-4|
• f(x)= |x| + 4
• f(x)= |x| - 4

Multiple Choice

Which function is graphed?

• f(x) = 2|x - 5|
• f(x) = |x - 5|
• f(x) = |x - 5| + 5
• f(x) = |x| + 5

Multiple Choice

Which function is graphed?

• f(x) = 2|x + 2| -4
• f(x) = 2|x - 4| + 2
• f(x) = |x + 2| - 4
• f(x) = -2|x - 4|

 $2\left|x-3\right|+4$  

• Multiplied by 2, the parent function becomes narrower. A vertical stretch.

• Subtracting inside the absolute value, we move right 3.

• Adding outside the absolute value, the graph translates up 4

• The vertex of the new function is located at (3, 4) Notice a pattern?

Multiple Choice

Identify the equation of the graph.

• $$f\left(x\right)=2\left(x-4\right)^2-5$$
  

• $$f\left(x\right)=\frac{1}{2}\left(x+4\right)^2-5$$
  

• $$f\left(x\right)=\frac{1}{2}\left(x-4\right)^2-5$$

• $$f\left(x\right)=2\left|x-4\right|-5$$

Multiple Choice

Identify the vertex of the function.

$$y=\left(x+1\right)^2+2$$   

• (1,2)

• (2,1)

• (-1,2)

• (-1,-2)

 $y=\left(x+1\right)^2+2$  

• Vertex shifts left one and up two from the origin.

• Vertex is (-1, 2)

• MORE ON QUADRATICS LATER!