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As with parallel lines, we can determine whether two lines are perpendicular by comparing their slopes, assuming that the lines are neither horizontal nor perpendicular. The slope of each line below is the negative reciprocal of the other so the lines are perpendicular.
The product of the slopes is –1.
Two lines are parallel lines if they do not intersect. The slopes of the lines are the same.
If and only if ${b}_{1}={b}_{2}$ and ${m}_{1}={m}_{2},$ we say the lines coincide. Coincident lines are the same line.
Two lines are perpendicular lines if they intersect at right angles.
Given the functions below, identify the functions whose graphs are a pair of parallel lines and a pair of perpendicular lines.
Parallel lines have the same slope. Because the functions $f(x)=2x+3$ and $j(x)=2x-6$ each have a slope of 2, they represent parallel lines. Perpendicular lines have negative reciprocal slopes. Because −2 and $\frac{1}{2}$ are negative reciprocals, the equations, $g(x)=\frac{1}{2}x-4$ and $h(x)=-2x+2$ represent perpendicular lines.
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.
Suppose for example, we are given the following equation.
We know that the slope of the line formed by the function is 3. We also know that the y- intercept is $\left(0,1\right).$ Any other line with a slope of 3 will be parallel to $f(x).$ So the lines formed by all of the following functions will be parallel to $f(x).$
Suppose then we want to write the equation of a line that is parallel to $f$ and passes through the point $\left(1,\text{7}\right).$ We already know that the slope is 3. We just need to determine which value for $b$ 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.
So $g(x)=3x+4$ is parallel to $f\left(x\right)=3x+1$ and passes through the point $\left(1,\text{7}\right).$
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.
Find a line parallel to the graph of $f(x)=3x+6$ that passes through the point $\left(3,\text{0}\right).$
The slope of the given line is 3. If we choose the slope-intercept form, we can substitute $m=3,$ $x=3,$ and $f(x)=0$ into the slope-intercept form to find the y- intercept.
The line parallel to $f(x)$ that passes through $\left(3,\text{0}\right)$ is $g(x)=3x-9.$
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