4.1 Linear functions  (Page 12/27)

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$-4\left(\frac{1}{4}\right)=-1$

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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