# 1.3 Rates of change and behavior of graphs  (Page 5/15)

 Page 5 / 15

Access this online resource for additional instruction and practice with rates of change.

## Key equations

 Average rate of change $\frac{\Delta y}{\Delta x}=\frac{f\left({x}_{2}\right)-f\left({x}_{1}\right)}{{x}_{2}-{x}_{1}}$

## Key concepts

• A rate of change relates a change in an output quantity to a change in an input quantity. The average rate of change is determined using only the beginning and ending data. See [link] .
• Identifying points that mark the interval on a graph can be used to find the average rate of change. See [link] .
• Comparing pairs of input and output values in a table can also be used to find the average rate of change. See [link] .
• An average rate of change can also be computed by determining the function values at the endpoints of an interval described by a formula. See [link] and [link] .
• The average rate of change can sometimes be determined as an expression. See [link] .
• A function is increasing where its rate of change is positive and decreasing where its rate of change is negative. See [link] .
• A local maximum is where a function changes from increasing to decreasing and has an output value larger (more positive or less negative) than output values at neighboring input values.
• A local minimum is where the function changes from decreasing to increasing (as the input increases) and has an output value smaller (more negative or less positive) than output values at neighboring input values.
• Minima and maxima are also called extrema.
• We can find local extrema from a graph. See [link] and [link] .
• The highest and lowest points on a graph indicate the maxima and minima. See [link] .

## Verbal

Can the average rate of change of a function be constant?

Yes, the average rate of change of all linear functions is constant.

If a function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is increasing on $\text{\hspace{0.17em}}\left(a,b\right)\text{\hspace{0.17em}}$ and decreasing on $\text{\hspace{0.17em}}\left(b,c\right),\text{\hspace{0.17em}}$ then what can be said about the local extremum of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}\left(a,c\right)?\text{\hspace{0.17em}}$

How are the absolute maximum and minimum similar to and different from the local extrema?

The absolute maximum and minimum relate to the entire graph, whereas the local extrema relate only to a specific region around an open interval.

How does the graph of the absolute value function compare to the graph of the quadratic function, $\text{\hspace{0.17em}}y={x}^{2},\text{\hspace{0.17em}}$ in terms of increasing and decreasing intervals?

## Algebraic

For the following exercises, find the average rate of change of each function on the interval specified for real numbers $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}h.$

$f\left(x\right)=4{x}^{2}-7\text{\hspace{0.17em}}$ on

$4\left(b+1\right)$

$g\left(x\right)=2{x}^{2}-9\text{\hspace{0.17em}}$ on

$p\left(x\right)=3x+4\text{\hspace{0.17em}}$ on

3

$k\left(x\right)=4x-2\text{\hspace{0.17em}}$ on

$f\left(x\right)=2{x}^{2}+1\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}\left[x,x+h\right]$

$4x+2h$

$g\left(x\right)=3{x}^{2}-2\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}\left[x,x+h\right]$

$a\left(t\right)=\frac{1}{t+4}\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}\left[9,9+h\right]$

$\frac{-1}{13\left(13+h\right)}$

$b\left(x\right)=\frac{1}{x+3}\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}\left[1,1+h\right]$

$j\left(x\right)=3{x}^{3}\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}\left[1,1+h\right]$

$3{h}^{2}+9h+9$

$r\left(t\right)=4{t}^{3}\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}\left[2,2+h\right]$

$\frac{f\left(x+h\right)-f\left(x\right)}{h}\text{\hspace{0.17em}}$ given $\text{\hspace{0.17em}}f\left(x\right)=2{x}^{2}-3x\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}\left[x,x+h\right]$

$4x+2h-3$

## Graphical

For the following exercises, consider the graph of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ shown in [link] .

Estimate the average rate of change from $\text{\hspace{0.17em}}x=1\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}x=4.$

Estimate the average rate of change from $\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}x=5.$

$\frac{4}{3}$

For the following exercises, use the graph of each function to estimate the intervals on which the function is increasing or decreasing.

increasing on $\text{\hspace{0.17em}}\left(-\infty ,-2.5\right)\cup \left(1,\infty \right),\text{\hspace{0.17em}}$ decreasing on

increasing on $\text{\hspace{0.17em}}\left(-\infty ,1\right)\cup \left(3,4\right),\text{\hspace{0.17em}}$ decreasing on $\text{\hspace{0.17em}}\left(1,3\right)\cup \left(4,\infty \right)$

For the following exercises, consider the graph shown in [link] .

#### Questions & Answers

The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
divide simplify each answer 3/2÷5/4
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
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