4.1 Linear functions  (Page 8/27)

The other characteristic of the linear function is its slope .

Let’s consider the following function.

The slope is $\frac{1}{2}.$ 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 $x=0.$ The graph crosses the y -axis at $(0,1).$ Now we know the slope and the y -intercept. We can begin graphing by plotting the point $(0,1).$ 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, $m=\frac{\text{rise}}{\text{run}}.$ From our example, we have $m=\frac{1}{2},$ which means that the rise is 1 and the run is 2. So starting from our y -intercept $(0,1),$ 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] .

Graphing by using the y- Intercept and slope

Graph $f(x)=-\frac{2}{3}x+5$ using the y- intercept and slope.

Evaluate the function at $x=0$ to find the y- intercept. The output value when $x=0$ is 5, so the graph will cross the y -axis at $(0,5).$

According to the equation for the function, the slope of the line is $-\frac{2}{3}.$ This tells us that for each vertical decrease in the “rise” of $–2$ 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 $(0,5)$ 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.

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Graphing a function using transformations

Another option for graphing is to use a transformation of the identity function $f(x)=x.$ 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.