Ch. 3 - Polynomial and Rational Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 31

Chapter 4, Problem 31

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=-2x(x-3)(x+2)

Verified step by step guidance

Identify the given polynomial function: $f(x) = -2x(x-3)(x+2)$. Notice that it is already factored into three linear factors multiplied together.

Determine the zeros of the function by setting each factor equal to zero: $x = 0$, $x - 3 = 0 \Rightarrow x = 3$, and $x + 2 = 0 \Rightarrow x = -2$. These are the x-intercepts of the graph.

Analyze the leading coefficient and degree to understand the end behavior. The leading term comes from multiplying $-2x$, $x$, $x-3$, and $x+2$. Since there are three linear factors and a coefficient of $-2$, the degree is 3 (odd degree) and the leading coefficient is negative, so the graph falls to the right and rises to the left.

Find the y-intercept by evaluating $f(0)$. Substitute $x=0$ into the function: $f(0) = -2 \times 0 \times (0-3) \times (0+2)$. This will give the point where the graph crosses the y-axis.

Plot the zeros and the y-intercept on the coordinate plane. Then, sketch the graph using the end behavior and the fact that the graph crosses the x-axis at the zeros. The graph will pass through each zero and change direction accordingly.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Factoring Polynomial Functions

Factoring involves expressing a polynomial as a product of its factors, which simplifies finding roots and analyzing the function. In this question, the polynomial is already factored as -2x(x-3)(x+2), making it easier to identify zeros and graph the function.

Zeros of a Polynomial Function

The zeros (or roots) of a polynomial are the values of x that make the function equal to zero. They correspond to the x-intercepts on the graph. For the given function, zeros occur where each factor equals zero: x = 0, x = 3, and x = -2.

Graphing Polynomial Functions

Graphing involves plotting key points such as zeros and determining the end behavior based on the leading coefficient and degree. The negative leading coefficient (-2) indicates the graph will fall to the right, and the multiplicity of roots affects whether the graph crosses or touches the x-axis at those points.

Recommended video:

05:25

Graphing Polynomial Functions

Related Practice

Textbook Question

Distance to the Horizon The distance that a person can see to the horizon on a clear day from a point above the surface of Earth varies directly as the square root of the height at that point. If a person 144 m above the surface of Earth can see 18 km to the horizon, how far can a person see to the horizon from a point 64 m above the surface?

511

views

Textbook Question

Use synthetic division to divide ƒ(x) by x-k for the given value of k. Then express ƒ(x) in the form ƒ(x) = (x-k) q(x) + r. ƒ(x) = 3x⁴ + 4x³ - 10x² + 15; k = -1

Factor $ƒ (x)$ into linear factors given that k is a zero. $ƒ (x) = x^{4} + 2 x^{3} - 7 x^{2} - 20 x - 12; k = - 2$ (multiplicity $2$ )

1010

views

Textbook Question

Graph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. ƒ(x) = x² + 6x + 5

997

views

Textbook Question

Match the rational function in Column I with the appropriate description in Column II. Choices in Column II can be used only once. ƒ(x)=1/(x+4)

871

views

Textbook Question

Solve each polynomial inequality. Give the solution set in interval notation. -(x - 3)(x - 4)² (x - 5) > 0

475

views