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The other characteristic of the linear function is its slope .
Let’s consider the following function.
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 $\text{\hspace{0.17em}}(0,1).\text{\hspace{0.17em}}$ Now we know the slope and the y -intercept. We can begin graphing by plotting the point $\text{\hspace{0.17em}}(0,1).\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}(0,1),$ 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] .
In the equation $\text{\hspace{0.17em}}f(x)=mx+b$
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.
Graph $\text{\hspace{0.17em}}f(x)=-\frac{2}{3}x+5\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}(0,5).$
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 $\text{\hspace{0.17em}}\u20132\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}(0,5)\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}(-3,7),\text{\hspace{0.17em}}$ $(-6,9),\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}(-9,11).$
Another option for graphing is to use a transformation of the identity function $\text{\hspace{0.17em}}f(x)=x.\text{\hspace{0.17em}}$ 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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