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The other characteristic of the linear function is its slope .

Let’s consider the following function.

f ( x ) = 1 2 x + 1

The slope is 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 = rise run . From our example, we have m = 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] .

This graph shows how to calculate the rise over run for the slope on an x, y coordinate plane.  The x-axis runs from negative 2 to 7. The y-axis runs from negative 2 to 5. The line extends right and upward from point (0,1), which is the y-intercept.  A dotted line extends two units to the right from point (0, 1) and is labeled Run = 2.  The same dotted line extends upwards one unit and is labeled Rise =1.

Graphical interpretation of a linear function

In the equation f ( x ) = m x + b

  • b is the y -intercept of the graph and indicates the point ( 0 , b ) at which the graph crosses the y -axis.
  • m 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:
m = change in output (rise) change in input (run) = Δ y Δ x = y 2 y 1 x 2 x 1

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 rise run 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 f ( x ) = 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 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.

This graph shows a decreasing function graphed on an x y coordinate plane. The x-axis runs from negative 3 to 7, and the y-axis runs from negative 1 to 7. The function passes through the points (0,5); (3,3); and (6,1).  Arrows extend downward two units and to the right three units from each point to the next point.
Graph of f ( x ) = −2 / 3 x + 5 and shows how to calculate the rise over run for the slope.
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Find a point on the graph we drew in [link] that has a negative x -value.

Possible answers include ( 3 , 7 ) , ( 6 , 9 ) , or ( 9 , 11 ) .

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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.

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