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(Section 10)

The appearance and position of a base equation from the Section 8 can be altered through transformations. Students preparing to take Calculus should be familiar with

Let's study a horizontal shift.   How will the functions y=f(x) and y=f(x+a) be similar and how will they be different?

Use the graph calculator to create the graphs and tables of y1 = x2 and y2 = (x+1)2.

graph graph
Graphs of y1 = x2 and y2 = (x+1)2 Table values of y1 = x2 and y2 = (x+1)2

Remember the base function y1 = x2 passes through (0,0), so y1 is the graph on the right.  What happens to the graph when x is replaced with the quantity x+1?  (The graph moves one unit to the left.)

Study the set of table values associated with each function to see how they are changed by replacing x with x+1.

(To obtain the same y values a value of x one less must be used. For example to obtain the value of 4 in y1 we use x = 2 or -2.  But to obtain a value of 4 in y2 we need to use 1 or -3.  These x values are one less.) 

Let's study a vertical stretch.  How will the functions y=f(x) and y=af(x) be similar and how will they be different?
Use the graphing calculator to look at graphs and tables of y1=|x| and y2=3|x|. 
graph graph
Graphs of y1=|x| and y2=3|x| Table values of y1=|x| and y2=3|x|

Remember the base function y1=|x| pass through the origin and the lines have slopes of 1 and -1.  What happens to the graph when the function is multiplied by 3? (The values of the function are multiplied by 3 or they become larger.)

Study the set of table values associated with each function to see how they are changed by multiplying by 3. 

(The y values from y1 are multiplied by 3 to get the y values for y2.  This means the y values will be higher for each x value.  So the graph of y=3|x| shows steeper lines.  No shift will take place, just a vertical stretch.) 

Let's study a reflection.  How will the graphs of y=f(x) and y=f(-x) be the same and how will they be different?
Use the graphing calculator to create graphs and tables of y1=3x and y2=3-x
graph table
Graphs of y1=3x and y2=3-x Table values of y1=3x and y2=3-x

Remember the base function  y1=3x begins in quadrant II and passes through the (0,1) and continues through quadrant I.  What happens to the graph when the x is replaced by -x.  (The graph is reflected over the y-axis.)

Study the set of table values associated with each function to see how they are changed by the replacement of x with -x.

(Studying the y values you can notice that y2(1)=y1(-1), y2(2)=y1(-2) or in general the table for y1 has been turned upside down to produce the table for y2.  This causes the graph to be reflected over the y-axis.)

Let's study a different reflection.  How will the graphs of y=f(x) and y=-f(x) be the same and how will they be different?
Use the graphing calculator to create graphs and tables of y1=3x and y2=-3x
graph table
Graphs of y1=3x and y2=-3x Table values of y1=3x and y2=-3x

Remember the base function y1=3x  begins in quadrant III and passes through the point (0,1) and continues through quadrant I.  What happens to the graph when the y=f(x) when it is changed to y=-f(x)?  (The graph is reflected over the x-axis.)

Study the set of table values associated with each function to see how the table values are changed by rewriting a function y=f(x) as y=-f(x). 

(The y values for y2 are opposite the y values for y1 for the same x values.  The graph has been reflected over the x-axis.)

Let's study a one more reflection.  How will the graphs of y=f(x) and y=-f(-x) be the same and how will they be different?
Use the graphing calculator to create graphs and tables of y1=ln(x) and y2=-ln(-x)
graph table
Graphs of y1=ln(x) and y2=-ln(-x) Table values of y1=ln(x) and y2=-ln(-x)

Remember the base function y1=ln(x) begins in quadrant IV and passes through the point (1,0) and continues through quadrant I.  What happens to the graph when the y=f(x) when it is changed to y=-f(-x)?  (There are two ways to look at the reflection:  1) The graph is reflected over the origin or 2) a double reflection has taken place-first over the y axis and then over the x axis.)

Study the set of table values associated with each function to see how the table values are changed by rewriting a function y=f(x) as y=-f(-x). 

(Studying the y values you can notice that y2(1)=-y1(-1), y2(2)=-y1(-2) or in general the table for y1 has been turned upside down to produce the table for y2 and the values have been changed to their opposite.  This causes the graph to be reflected over the origin or the double reflection.)

 

 

Summary

Vertical and horizontal shift transformations

Original Function Modified Function Change

If y = f(x)   

 y = f(x)+ a  shifts the function vertically a units
y = f(x)  y = f(x-a) shifts the function horizontally a units.
Stretching and shrinking transformations
Original Function Modified Function Change
y = f(x)  y = af(x) stretches the function vertically if a>1

y = f(x)

y = af(x) shrinks the function vertically if 0<a<1
Reflections
Original Function Modified Function Change

y = f(x) 

y = f(-x) reflect the function about the y-axis
y = f(x) y = -f(x)  reflect the function about the x-axis
y = f(x) y = -f(-x) reflect the function about the origin

Study the transformations demonstrated in this section. They will demonstrate the properties listed above. Note how the graphs are altered through the transformations.

Examples

1. equation (Refer to drawing I in Section 8)

Transformed Equations:

A)

graphequation

Note equation, so each point (x,y) on the original function is now transformed to (x, 3x+6).

Vertical shift: 6

Stretch: 3

No reflection

B)

graph

equaton

Note equation, so each point (x,y) on the original function is now transformed to (x, -6x+2). 

Vertical shift: 2

Stretch: 6

Reflection: about the x-axis

2. equation (Refer to graph III section 8) Vertex: (0,0)

Transformed Equations:

A)

graqph

equation

Horizontal shift: -2

No vertical shift

Vertex: (0,2)

No stretching or shrinking

No reflection

B)

graph
equation

No horizontal or vertical shift

Stretching factor: 3 (steeper)

C)

grah

equation

 

In this new function the x has be interchanged with y to create an equation which is no longer a function of x but instead a function of y. Note the change in the function.

D)

graph

equation

Horizontal shift: -3

Vertical Shift: 3

Vertex: (-3,3)

Shrinking factor: 0.1

Curvature: less steep

3.

graph

Center: (0,0)

Radius: 4

A)  Transformed Equation: equation

graph

(The reason the whole circle does not appear is because there is not pixel representing the point (-1,4) or (-9,4). If a different window had been selected, the circle might appear complete.) 

Vertical shift: 4

Horizontal shift: -5

Center: (-5,4)

Radius: unchanged

B) Transformed Equation: equation

graph

Vertical shift: 2

Horizontal shift: 3

Vertex: (3,2)

Radius: 5 (increased by 1)

4. equation (Refer to graph II in section 8)

graph

equation

Vertex: (0,0)

Equivalent piecewise equations:equation 

Transformed Equations:equation

A)

graph

Note B(x)=A(3x)

No horizontal or vertical shift

Stretching factor: 3

Two steeper lines

Piecewise equations: equation

B) Transformed equation: equation

graph

Horizontal shift: 2

Vertex: (2,0)

Piecewise Equations:equation

No stretching or shrinking 

C)  Transformed equation: equationor equation

graph

Horizontal shift: -1/2

Vertical Shift: 2

Vertex: (-1,2)

Stretching factor: 2 

Two steeper lines

Piecewise equations:equation

 

Check Your Understanding: (Remember to write your answers down to these questions.)

  1.  Describe what transformations have been performed on the base equationequation  when it is written as equationand then show the graph of this equation.

  2. Describe what transformations have been performed on the base equation equationwhen it is written as equationand then show the graph of this equation.

  3. Describe what transformations have been performed on the base equation  equtaion when it is written as .

  4. Describe what transformation have been performed on the base function y1 to produce the new set of table values for y2.

    table

  5. Describe what two transformations have been performed on the base function y1 to produce the new set of table values for y2.

    table

 

Table of Contents

Section 9

Answer Section

Section 11

 

This page was modified on 03/21/16

Rahn, 2000