# 2.1 Linear functions  (Page 9/17)

 Page 9 / 17

## Verbal

Terry is skiing down a steep hill. Terry's elevation, $E\left(t\right),$ in feet after $t$ seconds is given by $E\left(t\right)=3000-70t.$ Write a complete sentence describing Terry’s starting elevation and how it is changing over time.

Terry starts at an elevation of 3000 feet and descends 70 feet per second.

Maria is climbing a mountain. Maria's elevation, $E\left(t\right),$ in feet after $t$ minutes is given by $E\left(t\right)=1200+40t.$ Write a complete sentence describing Maria’s starting elevation and how it is changing over time.

Jessica is walking home from a friend’s house. After 2 minutes she is 1.4 miles from home. Twelve minutes after leaving, she is 0.9 miles from home. What is her rate in miles per hour?

3 miles per hour

Sonya is currently 10 miles from home and is walking farther away at 2 miles per hour. Write an equation for her distance from home t hours from now.

A boat is 100 miles away from the marina, sailing directly toward it at 10 miles per hour. Write an equation for the distance of the boat from the marina after t hours.

$d\left(t\right)=100-10t$

Timmy goes to the fair with $40. Each ride costs$2. How much money will he have left after riding $n$ rides?

## Algebraic

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

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

Yes.

$y=3x-5$

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

No.

$3x+5y=15$

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

No.

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

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

No.

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

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

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

Increasing.

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

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

Decreasing.

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

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

Decreasing.

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

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

Increasing.

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

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

Decreasing.

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

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

and

3

and

$\left(-1,\text{4}\right)$ and $\left(5,\text{2}\right)$

$–\frac{1}{3}$

$\left(8,-2\right)$ and $\left(4,6\right)$

and

$\frac{4}{5}$

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

$f\left(-5\right)=-4,$ and $f\left(5\right)=2$

$f\left(-1\right)=4$ and $f\left(5\right)=1$

$f\left(x\right)=-\frac{1}{2}x+\frac{7}{2}$

$\left(2,4\right)$ and $\left(4,10\right)$

Passes through $\left(1,5\right)$ and $\left(4,11\right)$

$y=2x+3$

Passes through and

Passes through and

$y=-\frac{1}{3}x+\frac{22}{3}$

x intercept at and y intercept at $\left(0,-3\right)$

x intercept at and y intercept at

$y=\frac{4}{5}x+4$

## Graphical

For the following exercises, find the slope of the lines graphed.

$-\frac{5}{4}$

For the following exercises, write an equation for the lines graphed.

$y=\frac{2}{3}x+1$

$y=-2x+3$

$y=3$

## Numeric

For the following exercises, which of the tables could represent a linear function? For each that could be linear, find a linear equation that models the data.

 $x$ 0 5 10 15 $g\left(x\right)$ 5 –10 –25 –40

Linear, $g\left(x\right)=-3x+5$

 $x$ 0 5 10 15 $h\left(x\right)$ 5 30 105 230
 $x$ 0 5 10 15 $f\left(x\right)$ –5 20 45 70

Linear, $f\left(x\right)=5x-5$

 $x$ 5 10 20 25 $k\left(x\right)$ 28 13 58 73
 $x$ 0 2 4 6 $g\left(x\right)$ 6 –19 –44 –69

Linear, $g\left(x\right)=-\frac{25}{2}x+6$

 $x$ 2 4 6 8 $f\left(x\right)$ –4 16 36 56
 $x$ 2 4 6 8 $f\left(x\right)$ –4 16 36 56

Linear, $f\left(x\right)=10x-24$

 $x$ 0 2 6 8 $k\left(x\right)$ 6 31 106 231

## Technology

If $f$ is a linear function, find an equation for the function.

$f\left(x\right)=-58x+17.3$

Graph the function $f$ on a domain of Enter the function in a graphing utility. For the viewing window, set the minimum value of $x$ to be $-10$ and the maximum value of $x$ to be $10.$

Graph the function $f$ on a domain of

#### Questions & Answers

how can are find the domain and range of a relations
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
i want to sure my answer of the exercise
what is the diameter of(x-2)²+(y-3)²=25
how to solve the Identity ?
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
what is a complex number used for?
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim