<< Chapter < Page Chapter >> Page >
y 4 = 3 4 ( x 4 ) y 4 = 3 4 x + 3 y = 3 4 x + 7

If we want to find the slope-intercept form without first writing the point-slope form, we could have recognized that the line crosses the y -axis when the output value is 7. Therefore, b = 7. We now have the initial value b and the slope m so we can substitute m and b into the slope-intercept form of a line.

This image shows the equation f of x equals m times x plus b. It shows that m is the value negative three fourths and b is 7. It then shows the equation rewritten as f of x equals negative three fourths times x plus 7.

So the function is f ( x ) = 3 4 x + 7 , and the linear equation would be y = 3 4 x + 7.

Given the graph of a linear function, write an equation to represent the function.

  1. Identify two points on the line.
  2. Use the two points to calculate the slope.
  3. Determine where the line crosses the y -axis to identify the y -intercept by visual inspection.
  4. Substitute the slope and y -intercept into the slope-intercept form of a line equation.

Writing an equation for a linear function

Write an equation for a linear function given a graph of f shown in [link] .

This figure shows an increasing function graphed on an x y coordinate plane. The x axis is labeled from negative 10 to 10. The y axis is labeled from negative 10 to 10. The function passes through the points (0, 2) and (-2, -4). These points are not labeled on this graph.

Identify two points on the line, such as ( 0 , 2 ) and ( −2 , −4 ) . Use the points to calculate the slope.

m = y 2 y 1 x 2 x 1 = −4 2 −2 0 = −6 −2 = 3

Substitute the slope and the coordinates of one of the points into the point-slope form.

y y 1 = m ( x x 1 ) y ( −4 ) = 3 ( x ( −2 ) ) y + 4 = 3 ( x + 2 )

We can use algebra to rewrite the equation in the slope-intercept form.

y + 4 = 3 ( x + 2 ) y + 4 = 3 x + 6 y = 3 x + 2
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Writing an equation for a linear cost function

Suppose Ben starts a company in which he incurs a fixed cost of $1,250 per month for the overhead, which includes his office rent. His production costs are $37.50 per item. Write a linear function C where C ( x ) is the cost for x items produced in a given month.

The fixed cost is present every month, $1,250. The costs that can vary include the cost to produce each item, which is $37.50. The variable cost, called the marginal cost, is represented by 37.5. The cost Ben incurs is the sum of these two costs, represented by C ( x ) = 1250 + 37.5 x .

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

Writing an equation for a linear function given two points

If f is a linear function, with f ( 3 ) = −2 , and f ( 8 ) = 1 , find an equation for the function in slope-intercept form.

We can write the given points using coordinates.

f ( 3 ) = 2 ( 3 , −2 ) f ( 8 ) = 1 ( 8 , 1 )

We can then use the points to calculate the slope.

m = y 2 y 1 x 2 x 1 = 1 ( −2 ) 8 3 = 3 5

Substitute the slope and the coordinates of one of the points into the point-slope form.

y y 1 = m ( x x 1 ) y ( −2 ) = 3 5 ( x 3 )

We can use algebra to rewrite the equation in the slope-intercept form.

y + 2 = 3 5 ( x 3 ) y + 2 = 3 5 x 9 5 y = 3 5 x 19 5
Got questions? Get instant answers now!
Got questions? Get instant answers now!

If f ( x ) is a linear function, with f ( 2 ) = –11 , and f ( 4 ) = −25 , write an equation for the function in slope-intercept form.

y = 7 x + 3

Got questions? Get instant answers now!

Modeling real-world problems with linear functions

In the real world, problems are not always explicitly stated in terms of a function or represented with a graph. Fortunately, we can analyze the problem by first representing it as a linear function and then interpreting the components of the function. As long as we know, or can figure out, the initial value and the rate of change of a linear function, we can solve many different kinds of real-world problems.

Given a linear function f and the initial value and rate of change, evaluate f ( c ) .

  1. Determine the initial value and the rate of change (slope).
  2. Substitute the values into f ( x ) = m x + b .
  3. Evaluate the function at x = c .

Questions & Answers

what are odd numbers
micheal Reply
numbers that leave a remainder when divided by 2
Thorben
1,3,5,7,... 99,...867
Thorben
7%2=1, 679%2=1, 866245%2=1
Thorben
the third and the seventh terms of a G.P are 81 and 16, find the first and fifth terms.
Suleiman Reply
if a=3, b =4 and c=5 find the six trigonometric value sin
Martin Reply
pls how do I factorize x⁴+x³-7x²-x+6=0
Gift Reply
in a function the input value is called
Rimsha Reply
how do I test for values on the number line
Modesta Reply
if a=4 b=4 then a+b=
Rimsha Reply
a+b+2ab
Kin
commulative principle
DIOSDADO
a+b= 4+4=8
Mimi
If a=4 and b=4 then we add the value of a and b i.e a+b=4+4=8.
Tariq
what are examples of natural number
sani Reply
an equation for the line that goes through the point (-1,12) and has a slope of 2,3
Katheryn Reply
3y=-9x+25
Ishaq
show that the set of natural numberdoes not from agroup with addition or multiplication butit forms aseni group with respect toaaddition as well as multiplication
Komal Reply
x^20+x^15+x^10+x^5/x^2+1
Urmila Reply
evaluate each algebraic expression. 2x+×_2 if ×=5
Sarch Reply
if the ratio of the root of ax+bx+c =0, show that (m+1)^2 ac =b^2m
Awe Reply
By the definition, is such that 0!=1.why?
Unikpel Reply
(1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin$)
Ajay Reply
hatdog
Mark
jaks
Ryan

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