# 4.1 Linear functions  (Page 8/27)

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The other characteristic of the linear function is its slope .

Let’s consider the following function.

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

The slope is $\text{\hspace{0.17em}}\frac{1}{2}.\text{\hspace{0.17em}}$ Because the slope is positive, we know the graph will slant upward from left to right. The y- intercept is the point on the graph when $\text{\hspace{0.17em}}x=0.\text{\hspace{0.17em}}$ The graph crosses the y -axis at $\text{\hspace{0.17em}}\left(0,1\right).\text{\hspace{0.17em}}$ Now we know the slope and the y -intercept. We can begin graphing by plotting the point $\text{\hspace{0.17em}}\left(0,1\right).\text{\hspace{0.17em}}$ We know that the slope is the change in the y -coordinate over the change in the x -coordinate. This is commonly referred to as rise over run, $\text{\hspace{0.17em}}m=\frac{\text{rise}}{\text{run}}.\text{\hspace{0.17em}}$ From our example, we have $\text{\hspace{0.17em}}m=\frac{1}{2},$ which means that the rise is 1 and the run is 2. So starting from our y -intercept $\text{\hspace{0.17em}}\left(0,1\right),$ we can rise 1 and then run 2, or run 2 and then rise 1. We repeat until we have a few points, and then we draw a line through the points as shown in [link] .

## Graphical interpretation of a linear function

In the equation $\text{\hspace{0.17em}}f\left(x\right)=mx+b$

• $b\text{\hspace{0.17em}}$ is the y -intercept of the graph and indicates the point $\text{\hspace{0.17em}}\left(0,b\right)\text{\hspace{0.17em}}$ at which the graph crosses the y -axis.
• $m\text{\hspace{0.17em}}$ is the slope of the line and indicates the vertical displacement (rise) and horizontal displacement (run) between each successive pair of points. Recall the formula for the slope:

Do all linear functions have y -intercepts?

Yes. All linear functions cross the y-axis and therefore have y-intercepts. (Note: A vertical line is parallel to the y-axis does not have a y-intercept, but it is not a function .)

Given the equation for a linear function, graph the function using the y -intercept and slope.

1. Evaluate the function at an input value of zero to find the y- intercept.
2. Identify the slope as the rate of change of the input value.
3. Plot the point represented by the y- intercept.
4. Use $\text{\hspace{0.17em}}\frac{\text{rise}}{\text{run}}\text{\hspace{0.17em}}$ to determine at least two more points on the line.
5. Sketch the line that passes through the points.

## Graphing by using the y- Intercept and slope

Graph $\text{\hspace{0.17em}}f\left(x\right)=-\frac{2}{3}x+5\text{\hspace{0.17em}}$ using the y- intercept and slope.

Evaluate the function at $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ to find the y- intercept. The output value when $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ is 5, so the graph will cross the y -axis at $\text{\hspace{0.17em}}\left(0,5\right).$

According to the equation for the function, the slope of the line is $\text{\hspace{0.17em}}-\frac{2}{3}.\text{\hspace{0.17em}}$ This tells us that for each vertical decrease in the “rise” of $\text{\hspace{0.17em}}–2\text{\hspace{0.17em}}$ units, the “run” increases by 3 units in the horizontal direction. We can now graph the function by first plotting the y -intercept on the graph in [link] . From the initial value $\text{\hspace{0.17em}}\left(0,5\right)\text{\hspace{0.17em}}$ we move down 2 units and to the right 3 units. We can extend the line to the left and right by repeating, and then drawing a line through the points.

Find a point on the graph we drew in [link] that has a negative x -value.

Possible answers include $\text{\hspace{0.17em}}\left(-3,7\right),\text{\hspace{0.17em}}$ $\left(-6,9\right),\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\left(-9,11\right).$

## Graphing a function using transformations

Another option for graphing is to use a transformation of the identity function $\text{\hspace{0.17em}}f\left(x\right)=x.\text{\hspace{0.17em}}$ A function may be transformed by a shift up, down, left, or right. A function may also be transformed using a reflection, stretch, or compression.

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