PCH; Ch 2.5 transformations of graphs

PCH: Transformations of graphs; Most of this should feel like a review

Family of Graphs

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You can see the PARENT graph in blue along with its equation. The purple graph has been shifted down 2 units and please see that the equation has a subtraction of 2 on the parent graph Finally, please compare the green graph and its equation to the purple graph and its equation.  Check out how the equations are different.

The vertex form of a quadratic function

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The information to your right is the basic idea for ALL functions of transformations.  

Multiple Choice

What is the best choice for the equation of this red graph?

• $$f\left(x\right)=2x^{ }$$

• $$f\left(x\right)=x^3$$

• $$f\left(x\right)=x^2$$

• $$f\left(x\right)=x$$

Multiple Select

This blue graph shifted up 3 units so the best choice for an equation would be?

• $$f\left(x\right)=x^2+3$$

• $$f\left(x\right)=x^2-3$$

• $$f\left(x\right)=x^2+1$$

Multiple Choice

Determine the best equation of this graph

• $$f\left(x\right)=x^2+4$$

• $$f\left(x\right)=x^2-4$$

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

Multiple Select

Choose from below the transformations of a parent graph of this function. $$f\left(x\right)=\left(x-4\right)^2$$  

• Take this parent graph  $$f\left(x\right)=x^2$$  and shift it to the left 4 units

• Take this parent graph $$f\left(x\right)=x^2$$  and shift it 4 units down

• Take this parent graph $$f\left(x\right)=x^2$$  and shift it 4 units to the right

Multiple Select

$$f\left(x\right)=\left(x+5\right)^2$$  

• $$f\left(x\right)=x^2\$$  gets shifted to the left 5 units.

• $$f\left(x\right)=x^2$$  gets shifted to the right 5 units 

• $$f\left(x\right)=x^2$$  gets shifted to up 5 units

Multiple Choice

$$f\left(x\right)=2x^2$$  takes the parent graph of $$f\left(x\right)=x^2$$  and.....

• vertically stretches it by a scale factor of 1/2

• vertically compresses it by a scale factor of 2

• vertically compresses it by a scale factor of 0.5

• vertically stretches it by a scale factor of 2

Multiple Choice

$$f\left(x\right)=\frac{1}{5}\left|x+3\right|$$  will take the function of  $$f\left(x\right)=\left|x+3\right|$$  and do which of the following...

• vertically compress by a scale factor of  $$\frac{1}{5}$$  

• vertically stretch by a scale factor of  $$\frac{1}{5}$$  

• vertically compress by a scale factor of 3

• vertically stretch by a scale factor of 3

​Other Functions are up next!

We will ask you to identify the graph from an equation and we will be transforming them.  

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Multiple Choice

What is the best choice for the equation of this graph?

• $$f\left(x\right)=x^2$$

• $$f\left(x\right)=x^3$$

• $$f\left(x\right)=\sqrt{x}$$

Multiple Choice

Which graph represents this equation?

Multiple Choice

Match the equation below with this graph.

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

• $$f\left(x\right)=x^3+2$$

• $$f\left(x\right)=x^3-2$$

Multiple Choice

Choose the equation for this graph

• $$f\left(x\right)=\left|x\right|$$  

• $$f\left(x\right)=x^2$$  

• $$f\left(x\right)=x^4$$  

Multiple Select

$$f\left(x\right)=\frac{1}{x}$$  matches with which graph?

You will see that this graph has been translated up 3 units and we would say reflected in the line y=3.  The reflection description with up/down shifting would be over the HORIZONTAL line of the constant value.  

Multiple Choice

Determine the best line of reflection for this graph

• $$y=-3$$

• $$x=-3$$

• the x-axis

• the y axis

Multiple Choice

Determine which equation below is the line of reflection for this graph (not the line of symmetry).

• $$y=2$$

• $$y=0$$

• $$x=2$$

• $$y=1$$

Square Root Family and Reflections

• Please observe the purple dotted line and the location of the negative sign in the equation.

• Please observe the red dotted line and the LOCATION of the negative sign in the equation.

• These are called reflections either over the y-axis or over the x-axis. 

Multiple Select

Be careful with this one....This equation matches with which graph?

$$f\left(x\right)=\sqrt{-x}+2$$  

Multiple Choice

Which equation below represents this graph?

• $$f\left(x\right)=\sqrt{-x+2}$$

• $$f\left(x\right)=-\sqrt{x}+2$$

• $$f\left(x\right)=-\sqrt{x+2}$$

Multiple Choice

How would you describe this reflection?

• reflected in the line x= -2

• reflected in the line y= - 2

• reflected in the x-axis

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HEY, DOWN HERE!!    Why does this equation look like it does NOT go with this graph?  What can be done to the equation to see the shift and the reflection?

​Factor out the negative inside the square root to get...

Multiple Choice

The equation  $$f\left(x\right)=\sqrt[]{-x+7}-3$$  would have what kind of transformations of $$f\left(x\right)=\sqrt[]{x}$$  ?

• Shift down 3 units then shift right 7 units and then reflect over the line x=7

• Shift down 3 units, then shift left 7 units and then reflect over the line y=7

• Shift up 3 units then shift left 7 units

• Shift up 3 units then shift right 7 units then reflect over the line y=3

Multiple Choice

$$f\left(x\right)=\sqrt{x+2}-3$$  this equation matches which graph below.  

Multiple Select

check all that apply to the following reciprocal function graph.

• Reciprocal Function

• $$f\left(x\right)=\frac{1}{x}$$

• $$f\left(x\right)=\left|x\right|$$

Multiple Choice

$$f\left(x\right)=\frac{1}{x}+2$$  

what do you think will happen to the reciprocal function if we do this....

• the parent graph will shift to the right 2 units

• the parent graph will shift to the left 2 units

• the parent graph will shift up 2 units

Multiple Choice

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

what will happen to the parent function after the equation has been changed to ....

• the graph will shift 1 unit right and 2 units down

• the graph will shift 1 unit left and 2 units down

• the graph will shift 2 units left and 1 unit down

Multiple Select

Be careful here!!! Check all that apply  $$f\left(x\right)=\frac{3}{x}$$  will take  $$f\left(x\right)=\frac{1}{x}$$  and.....

• vertically stretch the graph by a scale factor of 3

• vertically compress the graph by a scale factor of 3 

• horizontally stretch by a scale factor of 3

• horizontally compress by a scale factor of 3

Summary

• Identifying the parent function first is critical

• In general; horizontal shifts are values inside parentheses and they move the entire graph in the opposite direction of the sign.  (+ moves left, - moves right)

• In general; vertical shifts are values outside a parenthesis and they move the entire graph in the same direction on the y-axis than the sign that we see in the equation (+ moves up, - moves down)

• A negative in front of the parent function reflects over the horizontal line in that particular equation.

• A negative inside a parenthesis but part of the coefficient reflects the parent function over the vertical line  *these can be tricky*

• An 'a' value in front of the parent function vertically stretches or compresses depending on greater than one or between 0 and 1

• ​I noticed that square root graphs LOOK like the square root itself. Kind of.

Poll

Make one choice of how you feel the MOST about this activity

• I liked this

• It was okay

• meh

• just the textbook please

​Practice set is...

Pg 112/ 19-25 odds, 45, 49, 53, 57, 61, 65

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