6.08 Transformations of y=x

 

Transformations of $y=x$y=x

Previously, we discussed how the direct variation function $y=kx$y=kx forms the graph of a straight line.  The equation $y=kx$y=kx is one example of a transformation of the graph $y=x$y=x.  Now, explore other graphs that are within the same function family.

Guiding questions for translations

1. Move the slider for the $a$a value.  How would you describe the changes in the graph to a classmate?
2. Move the slider for the $b$b value.  How might you explain the changes to the graph to a classmate?
3. Given an equation of the form $y=ax+b$y=ax+b, how might you predict the graph of the function using your knowledge of transformations?

Guiding questions for rotations and reflections

1. Check the various boxes and notice what happens to the graph. Make an observation for each box.
2. Move around the blue point. Do you initial observations hold true?
3. Given an equation of the form $y=ax$y=ax, how might you predict the graph of the function using your knowledge of transformations?

 

Slope

So what you will have found is that the $a$a value affects the slope.

• If $a<0$a<0, the slope is negative and the line is decreasing
• if $a>0$a>0, the slope is positive and the line is increasing
• if $a=0$a=0 the slope is $0$0 and the line is horizontal
• Also, the larger the value of $a$a the steeper the line

$y$y-intercept

We also found that the $b$b value affects the $y$y intercept.  

• If $b$b is positive then the line is vertically translated (moved) up.
• If $b$b is negative then the line is vertically translated (moved) down.

 

So from equations in this form, $y=ax+b$y=ax+b, we instantly have enough information to understand what this line looks like and to describe the transformations from the basic line $y=x$y=x.

Worked examples

By first identifying the slope and $y$y intercept, describe the transformations of the following lines from the basic line $y=x$y=x.

Example 1

$y=3x$y=3x

• slope is $3$3
• $y$y intercept is $0$0
• Transformations of change: The line $y=x$y=x is made steeper due to a slope of $3$3 and is not vertically translated (it has the same $y$y-intercept as $y=x$y=x).

 

Example 2

$y=-2x$y=−2x

• slope is $-2$−2
• $y$y intercept is $0$0
• Transformations of change: The line $y=x$y=x is made steeper due to a slope of $2$2, is reflected on the $x$x-axis (due to a negative slope), and is not vertically translated.

 

Example 3

$y=\frac{x}{2}-3$y=x2​−3

• slope is $\frac{1}{2}$12​
• $y$y intercept is $-3$−3
• Transformations of change: the line $y=x$y=x is made less steep due to a slope of  $\frac{1}{2}$12​ and is vertically translated $3$3 units down (a $y$y-intercept of $-3$−3 compared to a $y$y-intercept of $0$0 in $y=x$y=x).

 

EXAMPLE 4

Consider the line $y=x$y=x shown below:

a) If the line was translated upwards by $2$2 units, sketch the graph of the new line and find the equation of the new line. 

Think: This translation would move the $y$y-intercept to the point $(0,2)$(0,2) but the line would keep the same level of steepness. The equation would only change by adding a constant of $2$2. 

Do: So the translated graph is shown below: 

And the equation of the new line would be: $y=x+2$y=x+2.

b) If the line $y=x$y=x was reflected across the $x$x-axis, sketch the graph of the new line and find the equation of the new line. 

Think: This reflection would cause the $y$y-coordinates of each point on the line to change sign, but the $x$x-coordinate would stay the same. So the point $(2,2)$(2,2) would be reflected to the point $(2,-2)$(2,−2).

Do: So the translated graph is shown below: 

And the equation of the new line is: $y=-x$y=−x.

c) If the line $y=x$y=x was rotated $90^\circ$90° clockwise, sketch the graph of the new line and find the equation of the new line. 

 

Think: This rotation would move the point $(3,3)$(3,3) to the point $(3,-3$(3,−3$)$). The angle between the line $y=x$y=x and either axis is $45^\circ$45°. This is also true for the line $y=-x$y=−x. So if we translate the line $y=x$y=x,  $45^\circ$45° clockwise, the line would lie on the $x$x-axis. Then if we rotate it another $45^\circ$45° clockwise it would like on the line $y=-x$y=−x.  

Do: So the translated graph is shown below: 

And the equation of the new line would be: $y=-x$y=−x.