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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.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

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 ) .

Got questions? Get instant answers now!

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.

Questions & Answers

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
Ken Reply
proof
AUSTINE
sebd me some questions about anything ill solve for yall
Manifoldee Reply
how to solve x²=2x+8 factorization?
Kristof Reply
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
SO THE ANSWER IS X=-8
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
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Prashant Reply
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Amit
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Dorbor
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Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
Reuben Reply
which of these functions is not uniformly cintinuous on (0, 1)? sinx
Pooja Reply
which of these functions is not uniformly continuous on 0,1
Basant Reply
solve this equation by completing the square 3x-4x-7=0
Jamiz Reply
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
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siame
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Jean Reply
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
Rubben Reply
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
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Nikki
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Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
Pab Reply
More example of algebra and trigo
Stephen Reply
What is Indices
Yashim Reply
If one side only of a triangle is given is it possible to solve for the unkown two sides?
Felix Reply
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Rubben
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Dayo Reply
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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