Q13. - Algebra 2

Question

Complete each of the following.

1. Graph each function by making a table of values.
2. Determine consecutive values of x between which each real zero is located.
3. Estimate the x-coordinate at which the relative maxima and relative minima occur.
   
   $f\left(x\right)=-x^3-4x^2$

Step-by-Step Solution

Answer

a) The graph of the polynomial function $f\left(x\right)=-x^3-4x^2$  is shown below:

b) The consecutive values of x between which each real zero of the function is located are -5 and -3, -1 and 1.

c) The estimated x-coordinate at which the relative maxima and relative minima occur for the function are 0 and -2.75 respectively.

a Step 1. Given information.

Given to graph the below function by making a table of values:

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

Step 2. Explanation.

Making a table of values of the function:

x	f(x)
-5	25
-4	0
-3	-9
-2	-8
-1	-3
0	0
1	-5

Graphing the function using the values in the table:

Step 3. Conclusion.

The graph of the given function is:

b Step 1. Given Information.

Given to determine the consecutive values of x between which each real zero of the function $f\left(x\right)=-x^3-4x^2$ is located.

Step 2. Explanation .

The real zero of a function lies where the value of the function changes from positive to negative or negative to positive.

From the table of values of the function, the value of the function changes two times between -5 and -3, -1 and 1.

Step 3. Conclusion .

Hence the consecutive values of x between which each real zero of the function is located are -5 and -3, -1 and 1.

c Step 1. Given Information.

Given to estimate the x-coordinate at which the relative maxima and relative minima occur for the function $f\left(x\right)=-x^3-4x^2$.

Step 2. Explanation .

In a graph, a relative maxima occurs when the graph changes the direction from upward to downward whereas a relative minima occurs when the graph changes the direction from downward to upward.

From the graph of the function, the relative maxima occurs at 0 and the relative minima occurs at -2.75 (approximately).

Step 3. Conclusion .

Hence the estimated x-coordinate at which the relative maxima and relative minima occur for the function are 0 and -2.75 respectively.