# 5.3 Graphs of polynomial functions  (Page 8/13)

 Page 8 / 13

Access the following online resource for additional instruction and practice with graphing polynomial functions.

## Key concepts

• Polynomial functions of degree 2 or more are smooth, continuous functions. See [link] .
• To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero. See [link] , [link] , and [link] .
• Another way to find the $\text{\hspace{0.17em}}x\text{-}$ intercepts of a polynomial function is to graph the function and identify the points at which the graph crosses the $\text{\hspace{0.17em}}x\text{-}$ axis. See [link] .
• The multiplicity of a zero determines how the graph behaves at the $\text{\hspace{0.17em}}x\text{-}$ intercepts. See [link] .
• The graph of a polynomial will cross the horizontal axis at a zero with odd multiplicity.
• The graph of a polynomial will touch the horizontal axis at a zero with even multiplicity.
• The end behavior of a polynomial function depends on the leading term.
• The graph of a polynomial function changes direction at its turning points.
• A polynomial function of degree $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ has at most $\text{\hspace{0.17em}}n-1\text{\hspace{0.17em}}$ turning points. See [link] .
• To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most $\text{\hspace{0.17em}}n-1\text{\hspace{0.17em}}$ turning points. See [link] and [link] .
• Graphing a polynomial function helps to estimate local and global extremas. See [link] .
• The Intermediate Value Theorem tells us that if have opposite signs, then there exists at least one value $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ between $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ for which $\text{\hspace{0.17em}}f\left(c\right)=0.\text{\hspace{0.17em}}$ See [link] .

## Verbal

What is the difference between an $\text{\hspace{0.17em}}x\text{-}$ intercept and a zero of a polynomial function $\text{\hspace{0.17em}}f?\text{\hspace{0.17em}}$

The $\text{\hspace{0.17em}}x\text{-}$ intercept is where the graph of the function crosses the $\text{\hspace{0.17em}}x\text{-}$ axis, and the zero of the function is the input value for which $\text{\hspace{0.17em}}f\left(x\right)=0.$

If a polynomial function of degree $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ has $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ distinct zeros, what do you know about the graph of the function?

Explain how the Intermediate Value Theorem can assist us in finding a zero of a function.

If we evaluate the function at $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and at $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ and the sign of the function value changes, then we know a zero exists between $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}b.$

Explain how the factored form of the polynomial helps us in graphing it.

If the graph of a polynomial just touches the x -axis and then changes direction, what can we conclude about the factored form of the polynomial?

There will be a factor raised to an even power.

## Algebraic

For the following exercises, find the $\text{\hspace{0.17em}}x\text{-}$ or t -intercepts of the polynomial functions.

$\text{\hspace{0.17em}}C\left(t\right)=2\left(t-4\right)\left(t+1\right)\left(t-6\right)$

$\text{\hspace{0.17em}}C\left(t\right)=3\left(t+2\right)\left(t-3\right)\left(t+5\right)$

$\left(-2,0\right),\left(3,0\right),\left(-5,0\right)$

$\text{\hspace{0.17em}}C\left(t\right)=4t{\left(t-2\right)}^{2}\left(t+1\right)$

$\text{\hspace{0.17em}}C\left(t\right)=2t\left(t-3\right){\left(t+1\right)}^{2}$

$\text{\hspace{0.17em}}\left(3,0\right),\left(-1,0\right),\left(0,0\right)$

$\text{\hspace{0.17em}}C\left(t\right)=2{t}^{4}-8{t}^{3}+6{t}^{2}$

$\text{\hspace{0.17em}}C\left(t\right)=4{t}^{4}+12{t}^{3}-40{t}^{2}$

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

$\text{\hspace{0.17em}}f\left(x\right)={x}^{3}+{x}^{2}-20x$

$f\left(x\right)={x}^{3}+6{x}^{2}-7x$

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

$f\left(x\right)={x}^{3}+2{x}^{2}-9x-18$

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

$\left(-2,0\right),\text{\hspace{0.17em}}\left(2,0\right),\text{\hspace{0.17em}}\left(\frac{1}{2},0\right)$

$f\left(x\right)={x}^{6}-7{x}^{3}-8$

$f\left(x\right)=2{x}^{4}+6{x}^{2}-8$

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

$f\left(x\right)={x}^{6}-2{x}^{4}-3{x}^{2}$

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

$f\left(x\right)={x}^{6}-3{x}^{4}-4{x}^{2}$

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

For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.

$f\left(x\right)={x}^{3}-9x,\text{\hspace{0.17em}}$ between $\text{\hspace{0.17em}}x=-4\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=-2.$

$f\left(x\right)={x}^{3}-9x,\text{\hspace{0.17em}}$ between $\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=4.$

$f\left(2\right)=–10\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(4\right)=28.$ Sign change confirms.

#### Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
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Sin(A+B) = sinBcosA+cosBsinA
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June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
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7.5 and 37.5
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find the sum of 28th term of the AP 3+10+17+---------
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I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
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Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
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