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olchik [2.2K]
4 years ago
12

NEED HELP EDU!! WILL GIVE BRAINLEIST! which graph of the cube root function f(x)=^3 √x?

Mathematics
1 answer:
Luden [163]4 years ago
5 0

Answer:

C. f(x)=\sqrt[3]{-x} -1

Step-by-step explanation:

Consider graph of the parent function (red curve in attached diagram)

g(x)=\sqrt[3]{x}

First, multiply it by -1 to get function

h(x)=-\sqrt[3]{x}

Then translate the graph of the function h(x) 1 unit down, then you'll get the function

f(x)=-\sqrt[3]{x} -1\\ \\ \text{or}\\ \\f(x)=\sqrt[3]{-x} -1

The graph of the function f(x) is represented by the blue curve in attached diagram

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Find a polynomial of degree 3 with real coefficients and zeros of -3,-1, and 4, for which f(-2)=-24
Zanzabum

We want to find a polynomial

<em>f(x)</em> = <em>a</em> <em>x</em>³ + <em>b</em> <em>x</em>² + <em>c</em> <em>x</em> + <em>d</em>

such that the roots of <em>f</em> are <em>x</em> = -3, <em>x</em> = -1, and <em>x</em> = 4, and <em>f(x)</em> takes on a value of -24 when <em>x</em> = -2.

The factor theorem for polynomials tells us that we can factorize <em>f(x)</em> as

<em>a</em> <em>x</em>³ + <em>b</em> <em>x</em>² + <em>c</em> <em>x</em> + <em>d</em> = <em>a</em> (<em>x</em> + 3) (<em>x</em> + 1) (<em>x</em> - 4)

Expand the right side:

(<em>x</em> + 3) (<em>x</em> + 1) (<em>x</em> - 4) = <em>x</em>³ - 13<em>x</em> - 12

So we have

<em>a</em> <em>x</em>³ + <em>b</em> <em>x</em>² + <em>c</em> <em>x</em> + <em>d</em> = <em>a x</em>³ - 13<em>a</em> <em>x</em> - 12<em>a</em>

<em />

In order for both sides to be equal, the coefficients of both polynomials on terms of the same degree must be equal. This means

<em>a</em> = <em>a</em> (of course)

<em>b</em> = 0 (there is no <em>x</em>² term on the right)

<em>c</em> = -13<em>a</em>

<em>d</em> = -12<em>a</em>

<em />

We also have that <em>f</em> (-2) = -24, which means

<em>f</em> (-2) = <em>a</em> (-2 + 3) (-2 + 1) (-2 - 4)

-24 = 6<em>a</em>

<em>a</em> = -4

which in turn tells us that <em>c</em> = 52 and <em>d</em> = 48.

So we found

<em>f(x)</em> = -4<em>x</em>³ + 52<em>x</em> + 48

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00.1

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