# 3.9 Modeling using variation  (Page 7/14)

 Page 7 / 14

$f\left(x\right)={x}^{5}+4{x}^{4}+4{x}^{3}$

$f\left(x\right)={x}^{3}-4{x}^{2}+x-4$

4 with multiplicity 1

For the following exercises, based on the given graph, determine the zeros of the function and note multiplicity.

$\frac{1}{2}\text{\hspace{0.17em}}$ with multiplicity 1, 3 with multiplicity 3

Use the Intermediate Value Theorem to show that at least one zero lies between 2 and 3 for the function $\text{\hspace{0.17em}}f\left(x\right)={x}^{3}-5x+1$

## Dividing Polynomials

For the following exercises, use long division to find the quotient and remainder.

$\frac{{x}^{3}-2{x}^{2}+4x+4}{x-2}$

$\text{\hspace{0.17em}}{x}^{2}+4\text{\hspace{0.17em}}$ with remainder 12

$\frac{3{x}^{4}-4{x}^{2}+4x+8}{x+1}$

For the following exercises, use synthetic division to find the quotient. If the divisor is a factor, then write the factored form.

$\frac{{x}^{3}-2{x}^{2}+5x-1}{x+3}$

${x}^{2}-5x+20-\frac{61}{x+3}$

$\frac{{x}^{3}+4x+10}{x-3}$

$\frac{2{x}^{3}+6{x}^{2}-11x-12}{x+4}$

$2{x}^{2}-2x-3$ , so factored form is $\left(x+4\right)\left(2{x}^{2}-2x-3\right)$

$\frac{3{x}^{4}+3{x}^{3}+2x+2}{x+1}$

## Zeros of Polynomial Functions

For the following exercises, use the Rational Zero Theorem to help you solve the polynomial equation.

$2{x}^{3}-3{x}^{2}-18x-8=0$

$3{x}^{3}+11{x}^{2}+8x-4=0$

$2{x}^{4}-17{x}^{3}+46{x}^{2}-43x+12=0$

$4{x}^{4}+8{x}^{3}+19{x}^{2}+32x+12=0$

For the following exercises, use Descartes’ Rule of Signs to find the possible number of positive and negative solutions.

${x}^{3}-3{x}^{2}-2x+4=0$

0 or 2 positive, 1 negative

$2{x}^{4}-{x}^{3}+4{x}^{2}-5x+1=0$

## Rational Functions

For the following rational functions, find the intercepts and the vertical and horizontal asymptotes, and then use them to sketch a graph.

$f\left(x\right)=\frac{x+2}{x-5}$

Intercepts $\left(–2,0\right)\text{and}\left(0,-\frac{2}{5}\right)$ , Asymptotes $\text{\hspace{0.17em}}x=5\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y=1.$

$f\left(x\right)=\frac{{x}^{2}+1}{{x}^{2}-4}$

$f\left(x\right)=\frac{3{x}^{2}-27}{{x}^{2}+x-2}$

Intercepts (3, 0), (-3, 0), and $\text{\hspace{0.17em}}\left(0,\frac{27}{2}\right)\text{\hspace{0.17em}}$ , Asymptotes

$f\left(x\right)=\frac{x+2}{{x}^{2}-9}$

For the following exercises, find the slant asymptote.

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

$f\left(x\right)=\frac{2{x}^{3}-{x}^{2}+4}{{x}^{2}+1}$

For the following exercises, find the inverse of the function with the domain given.

$f\left(x\right)={\left(x-2\right)}^{2},\text{\hspace{0.17em}}x\ge 2$

${f}^{-1}\left(x\right)=\sqrt{x}+2$

$f\left(x\right)={\left(x+4\right)}^{2}-3,\text{\hspace{0.17em}}x\ge -4$

$f\left(x\right)={x}^{2}+6x-2,\text{\hspace{0.17em}}x\ge -3$

${f}^{-1}\left(x\right)=\sqrt{x+11}-3$

$f\left(x\right)=2{x}^{3}-3$

$f\left(x\right)=\sqrt{4x+5}-3$

${f}^{-1}\left(x\right)=\frac{{\left(x+3\right)}^{2}-5}{4},\text{\hspace{0.17em}}x\ge -3$

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

## Modeling Using Variation

For the following exercises, find the unknown value.

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies directly as the square of $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ If when find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}x=4.$

$y=64$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies inversely as the square root of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ If when find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}x=4.$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies jointly as the cube of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and as $\text{\hspace{0.17em}}z.\text{\hspace{0.17em}}$ If when $\text{\hspace{0.17em}}x=1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z=2,\text{\hspace{0.17em}}$ $y=6,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z=3.$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies jointly as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and the square of $\text{\hspace{0.17em}}z\text{\hspace{0.17em}}$ and inversely as the cube of $\text{\hspace{0.17em}}w.\text{\hspace{0.17em}}$ If when $\text{\hspace{0.17em}}x=3,\text{\hspace{0.17em}}$ $z=4,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}w=2,\text{\hspace{0.17em}}$ $y=48,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}x=4,\text{\hspace{0.17em}}$ $z=5,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}w=3.$

For the following exercises, solve the application problem.

The weight of an object above the surface of the earth varies inversely with the distance from the center of the earth. If a person weighs 150 pounds when he is on the surface of the earth (3,960 miles from center), find the weight of the person if he is 20 miles above the surface.

148.5 pounds

The volume $\text{\hspace{0.17em}}V\text{\hspace{0.17em}}$ of an ideal gas varies directly with the temperature $\text{\hspace{0.17em}}T\text{\hspace{0.17em}}$ and inversely with the pressure P. A cylinder contains oxygen at a temperature of 310 degrees K and a pressure of 18 atmospheres in a volume of 120 liters. Find the pressure if the volume is decreased to 100 liters and the temperature is increased to 320 degrees K.

## Chapter test

Perform the indicated operation or solve the equation.

$\left(3-4i\right)\left(4+2i\right)$

$20-10i$

$\frac{1-4i}{3+4i}$

${x}^{2}-4x+13=0$

Give the degree and leading coefficient of the following polynomial function.

$f\left(x\right)={x}^{3}\left(3-6{x}^{2}-2{x}^{2}\right)$

Determine the end behavior of the polynomial function.

$f\left(x\right)=8{x}^{3}-3{x}^{2}+2x-4$

$As\text{\hspace{0.17em}}x\to -\infty ,\text{\hspace{0.17em}}f\left(x\right)\to -\infty ,\text{\hspace{0.17em}}as\text{\hspace{0.17em}}x\to \infty ,\text{\hspace{0.17em}}f\left(x\right)\to \infty$

$f\left(x\right)=-2{x}^{2}\left(4-3x-5{x}^{2}\right)$

Write the quadratic function in standard form. Determine the vertex and axes intercepts and graph the function.

$f\left(x\right)={x}^{2}+2x-8$

$f\left(x\right)={\left(x+1\right)}^{2}-9$ , vertex $\text{\hspace{0.17em}}\left(-1,-9\right)$ , intercepts $\text{\hspace{0.17em}}\left(2,0\right);\left(-4,0\right);\text{\hspace{0.17em}}\left(0,-8\right)$

Given information about the graph of a quadratic function, find its equation.

Vertex $\text{\hspace{0.17em}}\left(2,0\right)\text{\hspace{0.17em}}$ and point on graph $\text{\hspace{0.17em}}\left(4,12\right).$

Solve the following application problem.

A rectangular field is to be enclosed by fencing. In addition to the enclosing fence, another fence is to divide the field into two parts, running parallel to two sides. If 1,200 feet of fencing is available, find the maximum area that can be enclosed.

60,000 square feet

Find all zeros of the following polynomial functions, noting multiplicities.

$f\left(x\right)={\left(x-3\right)}^{3}\left(3x-1\right){\left(x-1\right)}^{2}$

$f\left(x\right)=2{x}^{6}-6{x}^{5}+18{x}^{4}$

0 with multiplicity 4, 3 with multiplicity 2

Based on the graph, determine the zeros of the function and multiplicities.

Use long division to find the quotient.

$\frac{2{x}^{3}+3x-4}{x+2}$

$2{x}^{2}-4x+11-\frac{26}{x+2}$

Use synthetic division to find the quotient. If the divisor is a factor, write the factored form.

$\frac{{x}^{4}+3{x}^{2}-4}{x-2}$

$\frac{2{x}^{3}+5{x}^{2}-7x-12}{x+3}$

$2{x}^{2}-x-4$ . So factored form is $\text{\hspace{0.17em}}\left(x+3\right)\left(2{x}^{2}-x-4\right)$

Use the Rational Zero Theorem to help you find the zeros of the polynomial functions.

$f\left(x\right)=2{x}^{3}+5{x}^{2}-6x-9$

$f\left(x\right)=4{x}^{4}+8{x}^{3}+21{x}^{2}+17x+4$

$-\frac{1}{2}\text{\hspace{0.17em}}$ (has multiplicity 2), $\text{\hspace{0.17em}}\frac{-1±i\sqrt{15}}{2}\text{\hspace{0.17em}}$

$f\left(x\right)=4{x}^{4}+16{x}^{3}+13{x}^{2}-15x-18$

$f\left(x\right)={x}^{5}+6{x}^{4}+13{x}^{3}+14{x}^{2}+12x+8$

$\text{\hspace{0.17em}}-2\text{\hspace{0.17em}}$ (has multiplicity 3), $\text{\hspace{0.17em}}±i$

Given the following information about a polynomial function, find the function.

It has a double zero at $\text{\hspace{0.17em}}x=3\text{\hspace{0.17em}}$ and zeroes at $\text{\hspace{0.17em}}x=1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=-2\text{\hspace{0.17em}}$ . It’s y -intercept is $\text{\hspace{0.17em}}\left(0,12\right).\text{\hspace{0.17em}}$

It has a zero of multiplicity 3 at $\text{\hspace{0.17em}}x=\frac{1}{2}\text{\hspace{0.17em}}$ and another zero at $\text{\hspace{0.17em}}x=-3\text{\hspace{0.17em}}$ . It contains the point $\text{\hspace{0.17em}}\left(1,8\right).$

$f\left(x\right)=2{\left(2x-1\right)}^{3}\left(x+3\right)$

Use Descartes’ Rule of Signs to determine the possible number of positive and negative solutions.

$8{x}^{3}-21{x}^{2}+6=0$

For the following rational functions, find the intercepts and horizontal and vertical asymptotes, and sketch a graph.

$f\left(x\right)=\frac{x+4}{{x}^{2}-2x-3}$

Intercepts $\text{\hspace{0.17em}}\left(-4,0\right),\text{\hspace{0.17em}}\left(0,-\frac{4}{3}\right)\text{\hspace{0.17em}}$ , Asymptotes .

$f\left(x\right)=\frac{{x}^{2}+2x-3}{{x}^{2}-4}$

Find the slant asymptote of the rational function.

$f\left(x\right)=\frac{{x}^{2}+3x-3}{x-1}$

$y=x+4$

Find the inverse of the function.

$f\left(x\right)=\sqrt{x-2}+4$

$f\left(x\right)=3{x}^{3}-4$

${f}^{-1}\left(x\right)=\sqrt[3]{\frac{x+4}{3}}$

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

Find the unknown value.

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies inversely as the square of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and when $\text{\hspace{0.17em}}x=3,\text{\hspace{0.17em}}$ $y=2.\text{\hspace{0.17em}}$ Find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}x=1.$

$y=18$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies jointly with $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and the cube root of $\text{\hspace{0.17em}}z.\text{\hspace{0.17em}}$ If when $\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z=27,\text{\hspace{0.17em}}$ $y=12,\text{\hspace{0.17em}}$ find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}x=5\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}z=8.$

Solve the following application problem.

The distance a body falls varies directly as the square of the time it falls. If an object falls 64 feet in 2 seconds, how long will it take to fall 256 feet?

4 seconds

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
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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
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what type of identity
Jeffrey
Confunction Identity
Barcenas
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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