# 4.1 Linear functions  (Page 15/27)

 Page 15 / 27

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 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.

## Algebraic

For the following exercises, determine whether the equation of the curve can be written as a linear function.

$y=\frac{1}{4}x+6$

$y=3x-5$

Yes

$y=3{x}^{2}-2$

$3x+5y=15$

Yes

$3{x}^{2}+5y=15$

$3x+5{y}^{2}=15$

No

$-2{x}^{2}+3{y}^{2}=6$

$-\frac{x-3}{5}=2y$

Yes

For the following exercises, determine whether each function is increasing or decreasing.

$f\left(x\right)=4x+3$

$g\left(x\right)=5x+6$

Increasing

$a\left(x\right)=5-2x$

$b\left(x\right)=8-3x$

Decreasing

$h\left(x\right)=-2x+4$

$k\left(x\right)=-4x+1$

Decreasing

$j\left(x\right)=\frac{1}{2}x-3$

$p\left(x\right)=\frac{1}{4}x-5$

Increasing

$n\left(x\right)=-\frac{1}{3}x-2$

$m\left(x\right)=-\frac{3}{8}x+3$

Decreasing

For the following exercises, find the slope of the line that passes through the two given points.

$\left(2,4\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(4,\text{10}\right)$

$\left(1,\text{5}\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(4,\text{11}\right)$

2

$\left(–1,\text{4}\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(5,\text{2}\right)$

$\left(8,–2\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(4,6\right)$

–2

$\left(6,11\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(–4,\text{3}\right)$

For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible.

$f\left(-5\right)=-4,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(5\right)=2$

$y=\frac{3}{5}x-1$

$f\left(-1\right)=4,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(5\right)=1$

Passes through $\text{\hspace{0.17em}}\left(2,4\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(4,10\right)$

$y=3x-2$

Passes through $\text{\hspace{0.17em}}\left(1,5\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(4,11\right)$

Passes through $\text{\hspace{0.17em}}\left(-1,\text{4}\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(5,\text{2}\right)$

$y=-\frac{1}{3}x+\frac{11}{3}$

Passes through $\text{\hspace{0.17em}}\left(-2,\text{8}\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(4,\text{6}\right)$

x intercept at $\text{\hspace{0.17em}}\left(-2,\text{0}\right)\text{\hspace{0.17em}}$ and y intercept at $\text{\hspace{0.17em}}\left(0,-3\right)$

$y=-1.5x-3$

x intercept at $\text{\hspace{0.17em}}\left(-5,\text{0}\right)\text{\hspace{0.17em}}$ and y intercept at $\text{\hspace{0.17em}}\left(0,\text{4}\right)$

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

$\begin{array}{c}3y+x=12\\ -y=8x+1\end{array}$

$\begin{array}{c}3y+4x=12\\ -6y=8x+1\end{array}$

parallel

$\begin{array}{l}6x-9y=10\hfill \\ 3x+2y=1\hfill \end{array}$

For the following exercises, find the x - and y- intercepts of each equation.

$f\left(x\right)=-x+2$

$\begin{array}{l}f\left(0\right)=-\left(0\right)+2\\ f\left(0\right)=2\\ y-\mathrm{int}:\left(0,2\right)\\ 0=-x+2\\ x-\mathrm{int}:\left(2,0\right)\end{array}$

$g\left(x\right)=2x+4$

$h\left(x\right)=3x-5$

$\begin{array}{l}h\left(0\right)=3\left(0\right)-5\\ h\left(0\right)=-5\\ y-\mathrm{int}:\left(0,-5\right)\\ 0=3x-5\\ x-\mathrm{int}:\left(\frac{5}{3},0\right)\end{array}$

$k\left(x\right)=-5x+1$

$-2x+5y=20$

$\begin{array}{l}-2x+5y=20\\ -2\left(0\right)+5y=20\\ 5y=20\\ y=4\\ y-\mathrm{int}:\left(0,4\right)\\ -2x+5\left(0\right)=20\\ x=-10\\ x-\mathrm{int}:\left(-10,0\right)\end{array}$

$7x+2y=56$

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 $\text{\hspace{0.17em}}\left(0,6\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(3,-24\right)$

Line 2: Passes through $\text{\hspace{0.17em}}\left(-1,19\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(8,-71\right)$

Line 1: m = –10 Line 2: m = –10 Parallel

Line 1: Passes through $\text{\hspace{0.17em}}\left(-8,-55\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(10,89\right)$

Line 2: Passes through $\text{\hspace{0.17em}}\left(9,-44\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(4,-14\right)$

Line 1: Passes through $\text{\hspace{0.17em}}\left(2,3\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(4,-1\right)$

Line 2: Passes through $\text{\hspace{0.17em}}\left(6,3\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(8,5\right)$

Line 1: m = –2 Line 2: m = 1 Neither

Line 1: Passes through $\text{\hspace{0.17em}}\left(1,7\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(5,5\right)$

Line 2: Passes through $\text{\hspace{0.17em}}\left(-1,-3\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(1,1\right)$

Line 1: Passes through $\text{\hspace{0.17em}}\left(2,5\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(5,-1\right)$

Line 2: Passes through $\text{\hspace{0.17em}}\left(-3,7\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(3,-5\right)$

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 $\text{\hspace{0.17em}}\left(2,\text{–}12\right).$

Write an equation for a line parallel to $\text{\hspace{0.17em}}g\left(x\right)=3x-1\text{\hspace{0.17em}}$ and passing through the point $\text{\hspace{0.17em}}\left(4,9\right).$

$y=3x-3$

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
proof
AUSTINE
sebd me some questions about anything ill solve for yall
how to solve x²=2x+8 factorization?
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
which of these functions is not uniformly cintinuous on (0, 1)? sinx
which of these functions is not uniformly continuous on 0,1
solve this equation by completing the square 3x-4x-7=0
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
More example of algebra and trigo
What is Indices
If one side only of a triangle is given is it possible to solve for the unkown two sides?
cool
Rubben
kya
Khushnama
please I need help in maths
Okey tell me, what's your problem is?
Navin