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2.2 Graphs of linear functions  (Page 2/15)

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Graph f ( x ) = − 3 4 x + 6 by plotting points.

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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 ( 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 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] .

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 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 draw a line through the points.

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Questions & Answers

for the "hiking" mix, there are 1,000 pieces in the mix, containing 390.8 g of fat, and 165 g of protein. if there is the same amount of almonds as cashews, how many of each item is in the trail mix?
ADNAN Reply
linear speed of an object
Melissa Reply
an object is traveling around a circle with a radius of 13 meters .if in 20 seconds a central angle of 1/7 Radian is swept out what are the linear and angular speed of the object
Melissa
test
Matrix
how to find domain
Mohamed Reply
like this: (2)/(2-x) the aim is to see what will not be compatible with this rational expression. If x= 0 then the fraction is undefined since we cannot divide by zero. Therefore, the domain consist of all real numbers except 2.
Dan
define the term of domain
Moha
if a>0 then the graph is concave
Angel Reply
if a<0 then the graph is concave blank
Angel
what's a domain
Kamogelo Reply
The set of all values you can use as input into a function su h that the output each time will be defined, meaningful and real.
Spiro
how fast can i understand functions without much difficulty
Joe Reply
what is inequalities
Nathaniel
functions can be understood without a lot of difficulty. Observe the following: f(2) 2x - x 2(2)-2= 2 now observe this: (2,f(2)) ( 2, -2) 2(-x)+2 = -2 -4+2=-2
Dan
what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
158.5 This number can be developed by using algebra and logarithms. Begin by moving log(2) to the right hand side of the equation like this: t/100 log(2)= log(3) step 1: divide each side by log(2) t/100=1.58496250072 step 2: multiply each side by 100 to isolate t. t=158.49
Dan
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
how to find x: 12x = 144 notice how 12 is being multiplied by x. Therefore division is needed to isolate x and whatever we do to one side of the equation we must do to the other. That develops this: x= 144/12 divide 144 by 12 to get x. addition: 12+x= 14 subtract 12 by each side. x =2
Dan
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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