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If the graphs of two linear functions are perpendicular, describe the relationship between the slopes and the y -intercepts.
If a horizontal line has the equation $\text{\hspace{0.17em}}f\left(x\right)=a\text{\hspace{0.17em}}$ and a vertical line has the equation $\text{\hspace{0.17em}}x=a,\text{\hspace{0.17em}}$ what is the point of intersection? Explain why what you found is the point of intersection.
The point of intersection is $\text{\hspace{0.17em}}\left(a,\text{}a\right).\text{\hspace{0.17em}}$ This is because for the horizontal line, all of the $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ coordinates are $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and for the vertical line, all of the $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ coordinates are $\text{\hspace{0.17em}}a.\text{\hspace{0.17em}}$ The point of intersection is on both lines and therefore will have these two characteristics.
For the following exercises, determine whether the equation of the curve can be written as a linear function.
$y=\frac{1}{4}x+6$
$y=3{x}^{2}-2$
$3{x}^{2}+5y=15$
$-2{x}^{2}+3{y}^{2}=6$
For the following exercises, determine whether each function is increasing or decreasing.
$f\left(x\right)=4x+3$
$a\left(x\right)=5-2x$
$h\left(x\right)=\mathrm{-2}x+4$
$j\left(x\right)=\frac{1}{2}x-3$
$n\left(x\right)=-\frac{1}{3}x-2$
For the following exercises, find the slope of the line that passes through the two given points.
$(2,4)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(4,\text{10})$
$(1,\text{5})\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(4,\text{11})$
2
$(\mathrm{\u20131},\text{4})\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(5,\text{2})$
$(8,\mathrm{\u20132})\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(4,6)$
–2
$(6,11)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(\mathrm{\u20134},\text{3})$
For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible.
$f(-5)=\mathrm{-4},\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f(5)=2$
$y=\frac{3}{5}x-1$
$f(\mathrm{-1})=4,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f(5)=1$
Passes through $\text{\hspace{0.17em}}(2,4)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(4,10)$
$y=3x-2$
Passes through $\text{\hspace{0.17em}}(1,5)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(4,11)$
Passes through $\text{\hspace{0.17em}}(\mathrm{-1},\text{4})\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(5,\text{2})$
$y=-\frac{1}{3}x+\frac{11}{3}$
Passes through $\text{\hspace{0.17em}}(\mathrm{-2},\text{8})\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(4,\text{6})$
x intercept at $\text{\hspace{0.17em}}(\mathrm{-2},\text{0})\text{\hspace{0.17em}}$ and y intercept at $\text{\hspace{0.17em}}(0,\mathrm{-3})$
$y=-1.5x-3$
x intercept at $\text{\hspace{0.17em}}(\mathrm{-5},\text{0})\text{\hspace{0.17em}}$ and y intercept at $\text{\hspace{0.17em}}(0,\text{4})$
For the following exercises, determine whether the lines given by the equations below are parallel, perpendicular, or neither.
$\begin{array}{l}4x-7y=10\hfill \\ 7x+4y=1\hfill \end{array}$
perpendicular
$\begin{array}{c}3y+x=12\\ -y=8x+1\end{array}$
$\begin{array}{l}6x-9y=10\hfill \\ 3x+2y=1\hfill \end{array}$
For the following exercises, find the x - and y- intercepts of each equation.
$f\left(x\right)=-x+2$
$\begin{array}{l}f(0)=-(0)+2\\ f(0)=2\\ y-\mathrm{int}:(0,2)\\ 0=-x+2\\ x-\mathrm{int}:(2,0)\end{array}$
$g\left(x\right)=2x+4$
$h\left(x\right)=3x-5$
$\begin{array}{l}h(0)=3(0)-5\\ h(0)=-5\\ y-\mathrm{int}:(0,-5)\\ 0=3x-5\\ x-\mathrm{int}:\left(\frac{5}{3},0\right)\end{array}$
$k\left(x\right)=\mathrm{-5}x+1$
$-2x+5y=20$
$\begin{array}{l}-2x+5y=20\\ -2(0)+5y=20\\ 5y=20\\ y=4\\ y-\mathrm{int}:(0,4)\\ -2x+5(0)=20\\ x=-10\\ x-\mathrm{int}:(-10,0)\end{array}$
$7x+2y=56$
For the following exercises, use the descriptions of each pair of lines given below to find the slopes of Line 1 and Line 2. Is each pair of lines parallel, perpendicular, or neither?
Line 1: Passes through $\text{\hspace{0.17em}}(0,6)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(3,\mathrm{-24})$
Line 2: Passes through $\text{\hspace{0.17em}}(\mathrm{-1},19)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(8,\mathrm{-71})$
Line 1: m = –10 Line 2: m = –10 Parallel
Line 1: Passes through $\text{\hspace{0.17em}}(\mathrm{-8},\mathrm{-55})\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(10,89)$
Line 2: Passes through $\text{\hspace{0.17em}}(9,-44)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(4,-14)$
Line 1: Passes through $\text{\hspace{0.17em}}(2,3)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(4,\mathrm{-1})$
Line 2: Passes through $\text{\hspace{0.17em}}(6,3)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(8,5)$
Line 1: m = –2 Line 2: m = 1 Neither
Line 1: Passes through $\text{\hspace{0.17em}}(1,7)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(5,5)$
Line 2: Passes through $\text{\hspace{0.17em}}(\mathrm{-1},\mathrm{-3})\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(1,1)$
Line 1: Passes through $\text{\hspace{0.17em}}(2,5)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(5,-1)$
Line 2: Passes through $\text{\hspace{0.17em}}(\mathrm{-3},7)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(3,\mathrm{-5})$
$\text{Line1}:m=\u20132\text{Line2}:m=\u20132\text{Parallel}$
For the following exercises, write an equation for the line described.
Write an equation for a line parallel to $\text{\hspace{0.17em}}f\left(x\right)=-5x-3\text{\hspace{0.17em}}$ and passing through the point $\text{\hspace{0.17em}}(2,\text{\u2013}12).$
Write an equation for a line parallel to $\text{\hspace{0.17em}}g(x)=3x-1\text{\hspace{0.17em}}$ and passing through the point $\text{\hspace{0.17em}}(4,9).$
$y=3x-3$
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