# 4.1 Linear functions  (Page 12/27)

 Page 12 / 27
$-4\left(\frac{1}{4}\right)=-1$

## Parallel and perpendicular lines

Two lines are parallel lines    if they do not intersect. The slopes of the lines are the same.

If and only if $\text{\hspace{0.17em}}{b}_{1}={b}_{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{m}_{1}={m}_{2},\text{\hspace{0.17em}}$ we say the lines coincide. Coincident lines are the same line.

Two lines are perpendicular lines    if they intersect to form a right angle.

${m}_{1}{m}_{2}=-1,\text{so}\text{\hspace{0.17em}}{m}_{2}=-\frac{1}{{m}_{1}}$

## Identifying parallel and perpendicular lines

Given the functions below, identify the functions whose graphs are a pair of parallel lines and a pair of perpendicular lines.

$\begin{array}{cccccc}\hfill f\left(x\right)& =& 2x+3\hfill & \hfill \phantom{\rule{2em}{0ex}}h\left(x\right)& =& -2x+2\hfill \\ \hfill g\left(x\right)& =& \frac{1}{2}x-4\hfill & \hfill \phantom{\rule{2em}{0ex}}j\left(x\right)& =& 2x-6\hfill \end{array}$

Parallel lines have the same slope. Because the functions $\text{\hspace{0.17em}}f\left(x\right)=2x+3\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}j\left(x\right)=2x-6\text{\hspace{0.17em}}$ each have a slope of 2, they represent parallel lines. Perpendicular lines have negative reciprocal slopes. Because −2 and $\text{\hspace{0.17em}}\frac{1}{2}\text{\hspace{0.17em}}$ are negative reciprocals, the functions $\text{\hspace{0.17em}}g\left(x\right)=\frac{1}{2}x-4\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}h\left(x\right)=-2x+2\text{\hspace{0.17em}}$ represent perpendicular lines.

## Writing the equation of a line parallel or perpendicular to a given line

If we know the equation of a line, we can use what we know about slope to write the equation of a line that is either parallel or perpendicular to the given line.

## Writing equations of parallel lines

Suppose for example, we are given the equation shown.

$f\left(x\right)=3x+1$

We know that the slope of the line formed by the function is 3. We also know that the y- intercept is $\text{\hspace{0.17em}}\left(0,1\right).\text{\hspace{0.17em}}$ Any other line with a slope of 3 will be parallel to $\text{\hspace{0.17em}}f\left(x\right).\text{\hspace{0.17em}}$ So the lines formed by all of the following functions will be parallel to $\text{\hspace{0.17em}}f\left(x\right).$

$\begin{array}{ccc}\hfill g\left(x\right)& =& 3x+6\hfill \\ \hfill h\left(x\right)& =& 3x+1\hfill \\ \hfill p\left(x\right)& =& 3x+\frac{2}{3}\hfill \end{array}$

Suppose then we want to write the equation of a line that is parallel to $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ and passes through the point $\text{\hspace{0.17em}}\left(1,\text{7}\right).\text{\hspace{0.17em}}$ This type of problem is often described as a point-slope problem because we have a point and a slope. In our example, we know that the slope is 3. We need to determine which value of $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ will give the correct line. We can begin with the point-slope form of an equation for a line, and then rewrite it in the slope-intercept form.

$\begin{array}{ccc}\hfill y-{y}_{1}& =& m\left(x-{x}_{1}\right)\hfill \\ \hfill y-7& =& 3\left(x-1\right)\hfill \\ \hfill y-7& =& 3x-3\hfill \\ \hfill y& =& 3x+4\hfill \end{array}$

So $\text{\hspace{0.17em}}g\left(x\right)=3x+4\text{\hspace{0.17em}}$ is parallel to $\text{\hspace{0.17em}}f\left(x\right)=3x+1\text{\hspace{0.17em}}$ and passes through the point $\text{\hspace{0.17em}}\left(1,\text{7}\right).$

Given the equation of a function and a point through which its graph passes, write the equation of a line parallel to the given line that passes through the given point.

1. Find the slope of the function.
2. Substitute the given values into either the general point-slope equation or the slope-intercept equation for a line.
3. Simplify.

## Finding a line parallel to a given line

Find a line parallel to the graph of $\text{\hspace{0.17em}}f\left(x\right)=3x+6\text{\hspace{0.17em}}$ that passes through the point $\text{\hspace{0.17em}}\left(3,\text{0}\right).$

The slope of the given line is 3. If we choose the slope-intercept form, we can substitute $\text{\hspace{0.17em}}m=3,x=3,$ and $\text{\hspace{0.17em}}f\left(x\right)=0\text{\hspace{0.17em}}$ into the slope-intercept form to find the y- intercept.

$\begin{array}{ccc}\hfill g\left(x\right)& =& 3x+b\hfill \\ \hfill 0& =& 3\left(3\right)+b\hfill \\ \hfill b& =& –9\hfill \end{array}$

The line parallel to $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ that passes through $\text{\hspace{0.17em}}\left(3,\text{0}\right)\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}g\left(x\right)=3x-9.$

## Writing equations of perpendicular lines

We can use a very similar process to write the equation for a line perpendicular to a given line. Instead of using the same slope, however, we use the negative reciprocal of the given slope. Suppose we are given the function shown.

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nothing up todat yet
Miranda
hi
jai
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jai
Miranda Drice
jai
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jai
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Miranda
I am living in india
jai
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Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
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jai
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jai
thanks
Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
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Jeffrey
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Miranda
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Jeffrey
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Miranda
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Jeffrey
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Miranda
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Miranda
Jeffrey
Jeffrey
Miranda
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Miranda
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Steve
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Steve
I don't know why. But Im trying to like it.
Jeffrey
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Jeffrey
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Miranda
what is the solution of the given equation?
which equation
Miranda
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Jeffrey
Miranda
Jeffrey
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
cos(- z)=cos z
Mustafa
what is a algebra
(x+x)3=?
6x
Obed
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
Hi
Abdel
hi
Ye
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Nokwanda
C'est comment
Abdel
Hi
Amanda
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SORIE
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Chinni
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Ranjay
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ANSHU
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Chinni
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Chinni
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Hassan
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SORIE
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Abdel
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Yaona
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SORIE
it's 12
what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
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
I don't understand how radicals works pls
How look for the general solution of a trig function