4.03 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
- Move the slider for the $m$m value. How would you describe the changes in the graph to a classmate?
- Move the slider for the $b$b value. How might you explain the changes to the graph to a classmate?
- Given an equation of the form $y=mx+b$y=mx+b, how might you predict the graph of the function using your knowledge of transformations?
Slope
So what you will have found is that the $m$m value affects the slope.
- If $m<0$m<0, the slope is negative and the line is decreasing
- if $m>0$m>0, the slope is positive and the line is increasing
- if $m=0$m=0 the slope is $0$0 and the line is horizontal
- Also, the larger the value of $m$m 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, $y=mx+b$y=mx+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).
Key features of linear functions
On the $xy$xy-plane, a straight line can be drawn in $4$4 ways. They can be in any direction and pass through any two points.
This means that straight lines can be:
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| decreasing | horizontal | ||
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| increasing | vertical |
- An increasing graph means that as $x$x values increase, the $y$y values increase.
- A decreasing graph means that as $x$x values increase, the $y$y values decrease.
- A horizontal graph means that as $x$x values change the $y$y values remain the same
- In a vertical graph the $x$x value is constant.
Regardless of all different shapes, all linear functions have some common characteristics.
Intercepts
The word intercept, in mathematics, refers to a point where a line, curve or function crosses or intersects with the axes.
- We can have $x$x-intercepts: where the line, curve or function crosses the $x$x-axis.
- We can have $y$y-intercepts: where the line, curve or function crosses the $y$y-axis.
Linear functions might have:
- an $x$x-intercept only (in the case of a vertical line)
- a $y$y-intercept only (in the case of horizontal lines)
- or most have $2$2 intercepts, both an $x$x-and a $y$y-intercept (in the case of increasing or decreasing functions)
End behavior
The end or extrema behavior of a function is a description of what happens past the viewing zone, what happens with the function outside of the area we can see.
Take this graph for example,
We can see in this graph $x$x values between $-5$−5 and $5$5, and $y$y values between $-1$−1 and $6$6. But we know that the graph goes on and on in the same linear fashion.
This is the end or extrema behavior. All linear functions share the same end behavior, with a linear function this extrema behavior is that the line continues forever in the same direction. No kinks, turns, or unexpected movement just continues in that direction forever.
Practice questions
Question 1
The lines $y=x+2$y=x+2 (black line) and $y=x$y=x (gray line) have been graphed on the same set of axes.
Is the line $y=x+2$y=x+2 increasing or decreasing?
Increasing
ADecreasing
BIs it steeper, flatter or equally steep as $y=x$y=x?
equally steep
Asteeper
Bflatter
C
Question 2
Consider the graph of $y=x$y=x. If this graph is translated $2$2 units down, which of the following would be the graph of the new line?
- Loading Graph...
A coordinate plane with an x-axis and a y-axis, each ranging from -6 to 6. A straight line crosses the plane, passing through the points $\left(0,2\right)$(0,2) and $\left(1,3\right)$(1,3). ALoading Graph...A coordinate plane with an x-axis and a y-axis, each ranging from -6 to 6. A straight line crosses the plane, passing through the points $\left(-2,0\right)$(−2,0) and $\left(0,-2\right)$(0,−2). BLoading Graph...A coordinate plane with an x-axis and a y-axis, each ranging from -6 to 6. A straight line crosses the plane, passing through the points $\left(2,0\right)$(2,0) and $\left(0,2\right)$(0,2). CLoading Graph...A coordinate plane with an x-axis and a y-axis, each ranging from -6 to 6. A straight line crosses the plane, passing through the points $\left(0,-2\right)$(0,−2) and coord$1,-1$1,−1. D
Question 3
Consider the graph of $y=x$y=x, with the point $\left(1,1\right)$(1,1) that lies on the graph. If the steepness of this graph is quadrupled, which of the following would be the graph of the new line?
- Loading Graph...
A graph of a function is plotted on a Coordinate Plane with an x-axis and y-axis, each ranging from -6 to 6. The graph crosses the plane, passing through the points $\left(0,0\right)$(0,0) and $\left(1,4\right)$(1,4). ALoading Graph...A graph of a function is plotted on a Coordinate Plane with an x-axis and y-axis, each ranging from -6 to 6. The graph crosses the plane, passing through the points $\left(0,0\right)$(0,0) and $\left(-1,4\right)$(−1,4). BLoading Graph...A graph of a function is plotted on a Coordinate Plane with an x-axis and y-axis, each ranging from -6 to 6. The graph crosses the plane, passing through the points $\left(0,4\right)$(0,4) and $\left(-4,0\right)$(−4,0). CLoading Graph...A graph of a function is plotted on a Coordinate Plane with an x-axis and y-axis, each ranging from -6 to 6. The graph crosses the plane, passing through the points $\left(0,4\right)$(0,4) and $\left(1,8\right)$(1,8). D
Horizontal and vertical lines
We can quickly identify that a line is horizontal if it is parallel to the $x$x-axis.
Similarly, a line is vertical if it's parallel to the $y$y-axis.
Horizontal lines
We can draw a set of points that makes a horizontal line in the same way we can draw any set of points as before. Consider the following table of values:
| $x$x | $-5$−5 | $-4$−4 | $-3$−3 | $-2$−2 |
|---|---|---|---|---|
| $y$y | $2$2 | $2$2 | $2$2 | $2$2 |
Plotting each column as points on the $xy$xy-plane gives us the following:
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| Plotting points from the table of values |
Clearly the set of points form a line that is parallel to the $x$x-axis. But just to confirm, we can draw a line through these points to show this.
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| Horizontal line passing through all four points |
Vertical lines
Just as we did before, we can plot a set of points obtained from a table of values and show that the points fall on a vertical line. Consider the following table of values:
| $x$x | $2$2 | $2$2 | $2$2 | $2$2 |
|---|---|---|---|---|
| $y$y | $-3$−3 | $-1$−1 | $1$1 | $3$3 |
Plotting the points in the table of values gives us the following graph. We can also pass a line through the points to show that the points lie on a vertical line.
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| Plotting points from the table of values |
Practice questions
Question 4
Plot the line $x=-8$x=−8 on the number plane.
- Loading Graph...
Question 5
Write down the equation of the graphed line.