1. Home
  2. Algebra and trigonometry
  3. Linear functions
  4. Linear functions

4.1 Linear functions  (Page 8/27)

<< Chapter < Page
  Algebra and trigonometry     Page 8 / 27
Chapter >> Page >

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

Discuss the differences between taste and flavor, including how other sensory inputs contribute to our  perception of flavor.
John Reply
taste refers to your understanding of the flavor . while flavor one The other hand is refers to sort of just a blend things.
Faith
While taste primarily relies on our taste buds, flavor involves a complex interplay between taste and aroma
Kamara
which drugs can we use for ulcers
Ummi Reply
omeprazole
Kamara
what
Renee
what is this
Renee
is a drug
Kamara
of anti-ulcer
Kamara
Omeprazole Cimetidine / Tagament For the complicated once ulcer - kit
Patrick
what is the function of lymphatic system
Nency Reply
Not really sure
Eli
to drain extracellular fluid all over the body.
asegid
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include: 1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
Adama
what is anatomy
Oyindarmola Reply
Anatomy is the identification and description of the structures of living things
Kamara
what's the difference between anatomy and physiology
Oyerinde Reply
Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
AI-Robot
what is enzymes all about?
Mohammed Reply
Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
Kamara
yes
Prince
how does the stomach protect itself from the damaging effects of HCl
Wulku Reply
little girl okay how does the stomach protect itself from the damaging effect of HCL
Wulku
it is because of the enzyme that the stomach produce that help the stomach from the damaging effect of HCL
Kamara
function of digestive system
Ali Reply
function of digestive
Ali
the diagram of the lungs
Adaeze Reply
what is the normal body temperature
Diya Reply
37 degrees selcius
Xolo
37°c
Stephanie
please why 37 degree selcius normal temperature
Mark
36.5
Simon
37°c
Iyogho
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature
Stephanie
37A c
Wulku
what is anaemia
Diya Reply
anaemia is the decrease in RBC count hemoglobin count and PVC count
Eniola
what is the pH of the vagina
Diya Reply
how does Lysin attack pathogens
Diya
acid
Mary
I information on anatomy position and digestive system and there enzyme
Elisha Reply
anatomy of the female external genitalia
Muhammad Reply
Organ Systems Of The Human Body (Continued) Organ Systems Of The Human Body (Continued)
Theophilus Reply
what's lochia albra
Kizito
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 10

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask