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Access these online resources for additional instruction and practice with graphs of linear functions.

Key concepts

  • Linear functions may be graphed by plotting points or by using the y -intercept and slope. See [link] and [link] .
  • Graphs of linear functions may be transformed by using shifts up, down, left, or right, as well as through stretches, compressions, and reflections. See [link] .
  • The y -intercept and slope of a line may be used to write the equation of a line.
  • The x -intercept is the point at which the graph of a linear function crosses the x -axis. See [link] and [link] .
  • Horizontal lines are written in the form, f ( x ) = b . See [link] .
  • Vertical lines are written in the form, x = b . See [link] .
  • Parallel lines have the same slope.
  • Perpendicular lines have negative reciprocal slopes, assuming neither is vertical. See [link] .
  • A line parallel to another line, passing through a given point, may be found by substituting the slope value of the line and the x - and y -values of the given point into the equation, f ( x ) = m x + b , and using the b that results. Similarly, the point-slope form of an equation can also be used. See [link] .
  • A line perpendicular to another line, passing through a given point, may be found in the same manner, with the exception of using the negative reciprocal slope. See [link] and [link] .
  • A system of linear equations may be solved setting the two equations equal to one another and solving for x . The y -value may be found by evaluating either one of the original equations using this x -value.
  • A system of linear equations may also be solved by finding the point of intersection on a graph. See [link] and [link] .

Section exercises

Verbal

If the graphs of two linear functions are parallel, describe the relationship between the slopes and the y -intercepts.

The slopes are equal; y -intercepts are not equal.

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If the graphs of two linear functions are perpendicular, describe the relationship between the slopes and the y -intercepts.

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If a horizontal line has the equation f ( x ) = a and a vertical line has the equation x = a , what is the point of intersection? Explain why what you found is the point of intersection.

The point of intersection is ( a , a ) . This is because for the horizontal line, all of the y coordinates are a and for the vertical line, all of the x coordinates are a . The point of intersection will have these two characteristics.

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Explain how to find a line parallel to a linear function that passes through a given point.

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Explain how to find a line perpendicular to a linear function that passes through a given point.

First, find the slope of the linear function. Then take the negative reciprocal of the slope; this is the slope of the perpendicular line. Substitute the slope of the perpendicular line and the coordinate of the given point into the equation y = m x + b and solve for b . Then write the equation of the line in the form y = m x + b by substituting in m and b .

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Algebraic

For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither parallel nor perpendicular:

4 x 7 y = 10 7 x + 4 y = 1

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Questions & Answers

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.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
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
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
hello
Jessica Reply
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
Karlee Reply
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
Jean Reply
rotation by 80 of (x^2/9)-(y^2/16)=1
Garrett Reply
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
bashiir Reply
what is the standard form if the focus is at (0,2) ?
Lorejean Reply
a²=4
Roy Reply
Practice Key Terms 5

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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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