# 2.2 Graphs of linear functions  (Page 2/15)

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Graph $f\left(x\right)=-\frac{3}{4}x+6$ by plotting points.

## Graphing a function using y- Intercept and slope

Another way to graph linear functions is by using specific characteristics of the function rather than plotting points. The first characteristic is its y- intercept, which is the point at which the input value is zero. To find the y- intercept , we can set $x=0$ in the equation.

The other characteristic of the linear function is its slope $m,$ which is a measure of its steepness. Recall that the slope is the rate of change of the function. The slope of a function is equal to the ratio of the change in outputs to the change in inputs. Another way to think about the slope is by dividing the vertical difference, or rise, by the horizontal difference, or run. We encountered both the y- intercept and the slope in Linear Functions .

Let’s consider the following function.

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

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 $\left(0,1\right).$ Now we know the slope and the y -intercept. We can begin graphing by plotting the point $\left(0,1\right)$ We know that the slope is 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 $\left(0,1\right),$ 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 $f\left(x\right)=mx+b$

• $b$ is the y -intercept of the graph and indicates the point $\left(0,b\right)$ 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:

Do all linear functions have y -intercepts?

Yes. All linear functions cross the y-axis and therefore have y-intercepts. (Note: A vertical line 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 $\frac{\text{rise}}{\text{run}}$ 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 $f\left(x\right)=-\frac{2}{3}x+5$ 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 $\left(0,5\right).$

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 $\left(0,5\right)$ 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 draw a line through the points.

how can are find the domain and range of a relations
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
i want to sure my answer of the exercise
what is the diameter of(x-2)²+(y-3)²=25
how to solve the Identity ?
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
what is a complex number used for?
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim