Question

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Explain how to sketch a graph of the function f(x) = x3 + 2x2 – 8x. Be sure to include end-behavior, zeroes, and intervals where the function is positive and negative.​

Answer 1

First, for end behavior, the highest power of x is x^3 and it is positive. So towards infinity, the graph will be positive, and towards negative infinity the graph will be negative (because this is a cubic graph)

To find the zeros, you set the equation equal to 0 and solve for x

x^3+2x^2-8x=0

x(x^2+2x-8)=0

x(x+4)(x-2)=0

x=0 x=-4 x=2

So the zeros are at 0, -4, and 2. Therefore, you can plot the points (0,0), (-4,0) and (2,0)

And we can plug values into the original that are between each of the zeros to see which intervals are positive or negative.

Plugging in a -5 gets us -35

-1 gets us 9

1 gets us -5

3 gets us 21

So now you know end behavior, zeroes, and signs of intervals

Hope this helps

Answer 2

The degree of the function is odd and the leading coefficient is positive – so the function goes to negative infinity as x goes to negative infinity and to positive infinity as x goes to positive infinity. The zeroes are –4, 0, and 2, all with multiplicity 1. The function is negative from negative infinity to –4 and from 0 to 2. The function is positive from –4 to 0 and from 2 to infinity.

Explanation:

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