Problem 14 Sketch the graph of the polynomi... [FREE SOLUTION]

Chapter 4: Problem 14

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior. $$ P(x)=(2 x-1)(x+1)(x+3) $$

Short Answer

Expert verified

Draw the graph with x-intercepts at $-3, -1, \frac{1}{2}$, a y-intercept at $-3$, and end behavior rising to the right and falling to the left.

Step by step solution

Identify the roots of the polynomial

The polynomial is given in factored form: $ P(x) = (2x-1)(x+1)(x+3) $. We can find the roots by setting each factor equal to zero. This gives roots at $ x = \frac{1}{2}, x = -1, \text{ and } x = -3 $. These are the x-intercepts of the graph.

Determine the y-intercept

To find the y-intercept, substitute $ x = 0 $ into the polynomial. \[ P(0) = (2(0) - 1)(0 + 1)(0 + 3) = (-1)(1)(3) = -3 \]. So, the y-intercept is $ (0, -3) $.

Analyze the end behavior

The polynomial $ P(x) = (2x-1)(x+1)(x+3) $ has a degree of 3, since there are three linear factors. The leading term is $ 2x^3 $. As $ x \to \infty $, $ P(x) \to \, \infty $ because a positive cubic term increases. As $ x \to -\infty $, $ P(x) \to \, -\infty $, because the leading term remains negative for large negative $ x $.

Sketch the graph

Using the roots $ x = \frac{1}{2}, x = -1, \text{ and } x = -3 $ and the y-intercept $ (0, -3) $, sketch the graph starting from the bottom right (since it rises to the right) and passing through these points. The curve should start from $ -\infty $ as $ x \to -\infty $, rise through $ x = -3 $, dip through $ x = -1 $, rise again through $ x = \frac{1}{2} $, and continue upward as $ x \to \infty $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Roots

The roots of a polynomial are the values of $x$ where the polynomial equals zero. These are also known as the x-intercepts of the graph because they occur at points where the graph crosses the x-axis. In the case of our polynomial $P(x) = (2x-1)(x+1)(x+3)$, setting each factor equal to zero allows us to easily find these roots. This gives us roots at $x = \frac{1}{2}$, $x = -1$, and $x = -3$.
To find the roots, consider solving these simple equations:

For $2x-1=0$, solve to get $x = \frac{1}{2}$.
For $x+1=0$, solve to get $x = -1$.
For $x+3=0$, solve to get $x = -3$.

At these x-values, the polynomial graph will touch or cross the x-axis.

Intercepts

Intercepts play a crucial role in the graph of a polynomial function, signaling the points where the graph meets the axes. Besides the x-intercepts (roots), another key intercept is the y-intercept. This is the point where the graph crosses the y-axis. To find the y-intercept of a polynomial, we evaluate the polynomial at $x = 0$.
Substituting $x = 0$ in $P(x)$:
\[ P(0) = (2 \times 0 - 1)(0 + 1)(0 + 3) = -1 \times 1 \times 3 = -3 \]
This calculation shows the y-intercept is $(0, -3)$.
Thus, our polynomial graph will cross the y-axis at the point $(0, -3)$. Remember, intercepts easily guide you towards sketching an accurate graph.

End Behavior

The end behavior of a graph describes how the graph acts as $x$ approaches infinity or negative infinity. For any polynomial, this is determined by its leading term when fully expanded.
In $P(x) = (2x-1)(x+1)(x+3)$, the degree of the polynomial is 3, because there are three linear factors. The leading term after multiplying these factors is $2x^3$.
Following these guidelines:

If the leading coefficient is positive and the degree is odd, like our $2x^3$, the graph rises to the right and falls to the left.
As $x \to \infty$, $2x^3 \to \infty$
As $x \to -\infty$, $2x^3 \to -\infty$

This pattern is what guides the overall shape and direction of the polynomial graph.

Factored Form

Understanding the factored form of a polynomial is crucial as it provides direct insight into the roots of the polynomial. Factored form is an expression where the polynomial is written as a product of its linear factors.
For $P(x) = (2x-1)(x+1)(x+3)$, we already have it in factored form. Each factor $(2x-1)$, $(x+1)$, and $(x+3)$ represents a simpler expression we can solve for to find the polynomial's roots.
Factored form offers these benefits:

Immediate identification of x-intercepts at solutions of each factor (discussed in polynomial roots).
Helpful for sketching polynomial graphs, as knowing where roots and intercepts are is crucial.
It simplifies analyzing the polynomial without the need to perform lengthy algebraic expansions.

Remember, writing polynomials in factored form makes graphing, evaluating, and understanding their behavior much easier.

91影视

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior. $$ P(x)=(2 x-1)(x+1)(x+3) $$

Short Answer

Step by step solution

Identify the roots of the polynomial

Determine the y-intercept

Analyze the end behavior

Sketch the graph

Key Concepts

Polynomial Roots

Intercepts

End Behavior

Factored Form

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