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.
So what you will have found is that the $a$a value affects the slope.
We also found that the $b$b value affects the $y$y intercept.
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.
By first identifying the slope and $y$y intercept, describe the transformations of the following lines from the basic line $y=x$y=x.
$y=3x$y=3x
$y=-2x$y=−2x
$y=\frac{x}{2}-3$y=x2−3
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.
Translations: when we slide the graph up, down, left, or right
Reflections: when we reflect the graph across one of the axes
Rotations: when we rotate the graph about the origin
Rotating $y=ax$y=ax clockwise $180^\circ$180° or counter-clockwise $180^\circ$180° gives $y=ax$y=ax