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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
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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 .

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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
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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

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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

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Steve
cosx.cos2x.cos4x.cos8x
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sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
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sinx sin2x is linearly dependent
cr
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The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
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cosA\1+sinA=secA-tanA
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The numbers cannot be combined with the x
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2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
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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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