Chapter 4: Q. 83 (page 210)

Analyze each polynomial function using Steps 1 through 8 on page 190 in Section 4.1
$f (x) = 4 x^{5} - 8 x^{4} - x + 2$

Short Answer

Expert verified

For the given polynomial function, we get:

x-intercept $= - \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}, 2$

y-intercept $= 2$

Step by step solution

Step 1. End behavior of the function

We have a degree of five from $f (x) = 4 x^{5} - 8 x^{4} - x + 2$ . Thus, the graph of function has an end behavior similar to $y = 4 x^{5}$

Step 2. Determine the intercept of the function

Determine the intercept of function to get the x- intercept.

$\begin{array}{rr} 4 x^{5} - 8 x^{4} - x + 2 = 0 \\ 4 x^{4} (x - 2) - (x - 2) = 0 \\ (4 x^{4} - 1) (x - 2) = 0 \\ 2 x^{2} + 1 = 0; 2 x^{2} - 1 = 0 x - 2 = 0 \\ 2 x^{2} = - 1; 2 x^{2} = 1; x = 2 \\ \sqrt{x^{2}} = \sqrt{- \frac{1}{2};} \sqrt{x^{2}} = \sqrt{\frac{1}{2}} \\ x = \frac{\sqrt{2}}{2}; x = - \frac{\sqrt{2}}{2}; \end{array}$

Considering $f (0) = 2$ , thus y-intercept is $(0, 2)$

Step 3. Graph the function

Step 4. Maximum and minimum turning point

From the graph above, we can see that the maximum and minimum turning point of the function is:

$(- 0.298, 2.226)$ and $(1.612, - 10.092)$

Step 5. Graph of function with obtained result

Step 6. Domain and range

Therefore, the domain and range of the function is the set of all real numbers.

Step 7. Interval at which the graph is increasing or decreasing

By observing the graph above, we say that the function at the intervals of $(- \infty, - 0.298)$ and $(1.612, \infty)$ is increasing.

At intervals of $(- 0.298, 1.612)$ decreasing.

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Analyze each polynomial function using Steps 1 through 8 on page 190 in Section 4.1
$f (x) = 4 x^{5} - 8 x^{4} - x + 2$

Short Answer

Step by step solution

Step 1. End behavior of the function

Step 2. Determine the intercept of the function

Step 3. Graph the function

Step 4. Maximum and minimum turning point

Step 5. Graph of function with obtained result

Step 6. Domain and range

Step 7. Interval at which the graph is increasing or decreasing

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Analyze each polynomial function using Steps 1 through 8 on page 190 in Section 4.1f(x)=4x5-8x4-x+2

Short Answer

Step by step solution

Step 1. End behavior of the function

Step 2. Determine the intercept of the function

Step 3. Graph the function

Step 4. Maximum and minimum turning point

Step 5. Graph of function with obtained result

Step 6. Domain and range

Step 7. Interval at which the graph is increasing or decreasing

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

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Statistics

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Logic and Functions

Study anywhere. Anytime. Across all devices.

Analyze each polynomial function using Steps 1 through 8 on page 190 in Section 4.1
$f (x) = 4 x^{5} - 8 x^{4} - x + 2$