# 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 $\text{\hspace{0.17em}}\left(4,9\right).$

$y=3x-3$

#### Questions & Answers

Cos45/sec30+cosec30=
Cos 45 = 1/ √ 2 sec 30 = 2/√3 cosec 30 = 2. =1/√2 / 2/√3+2 =1/√2/2+2√3/√3 =1/√2*√3/2+2√3 =√3/√2(2+2√3) =√3/2√2+2√6 --------- (1) =√3 (2√6-2√2)/((2√6)+2√2))(2√6-2√2) =2√3(√6-√2)/(2√6)²-(2√2)² =2√3(√6-√2)/24-8 =2√3(√6-√2)/16 =√18-√16/8 =3√2-√6/8 ----------(2)
exercise 1.2 solution b....isnt it lacking
I dnt get dis work well
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what is the procedure in solving quadratic equetion at least 6?
Almighty formula or by factorization...or by graphical analysis
Damian
I need to learn this trigonometry from A level.. can anyone help here?
yes am hia
Miiro
tanh2x =2tanhx/1+tanh^2x
cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)=cotb ... pls some one should help me with this..thanks in anticipation
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
cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)= cotb
favour
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
SO THE ANSWER IS X=-8
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
i am in
Cliff
1KI POWER 1/3 PLEASE SOLUTIONS
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Dorbor
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Find the possible value of 8.5 using moivre's theorem
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helo
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which of these functions is not uniformly continuous on 0,1