Problem 11 - Algebra for College Students

Question

For Problems \(1-22\), graph each of the polynomial functions. $$ f(x)=(x-2)(x+1)(x+3) $$

Step-by-Step Solution

Answer

Graph intersects x-axis at -3, -1, and 2 with ends going to negative and positive infinity.

Step 1: Identify Zeros of the Function

The polynomial function is given as \( f(x) = (x-2)(x+1)(x+3) \). To find the zeros, set each factor equal to zero. This gives \( x-2=0 \), \( x+1=0 \), and \( x+3=0 \). Solving these equations, the zeros are \( x=2 \), \( x=-1 \), and \( x=-3 \). These zeros are the points where the graph will intersect the x-axis.

Step 2: Determine the End Behavior

Since the polynomial is of degree 3 (the highest power of \( x \) is 3), we know that the graph will have a cubic shape. The leading term when expanded will be \( x^3 \), which has a positive coefficient. Therefore, the end behavior of the graph is that as \( x \to \infty \), \( f(x) \to \infty \) and as \( x \to -\infty \), \( f(x) \to -\infty \).

Step 3: Plot the Zeros on the Graph

On the coordinate plane, mark the x-intercepts at \( x = 2 \), \( x = -1 \), and \( x = -3 \). These are the points where the graph will cross the x-axis.

Step 4: Sketch the Graph Using Zeros and End Behavior

Using the information from Steps 1 and 2, sketch a smooth curve that passes through the x-intercepts at \( x = 2 \), \( x = -1 \), and \( x = -3 \). Start from the bottom left (since as \( x \to -\infty \), \( f(x) \to -\infty \)), pass through \( x = -3 \), turning around near \( x = -1 \), and finally reaching upward as \( x \to \infty \), crossing at \( x = 2 \). Ensure that the graph reflects the cubic behavior, being smooth and continuous.

Key Concepts

Zeros of Polynomial

Finding the zeros of a polynomial is crucial because these values show where the graph of the function crosses the x-axis. The polynomial function given is \[ f(x) = (x-2)(x+1)(x+3) \]. To find the zeros, we set each factor equal to zero:

• For \( x-2=0 \), solving gives \( x=2 \).
• For \( x+1=0 \), solving gives \( x=-1 \).
• For \( x+3=0 \), solving gives \( x=-3 \).

These zeros indicate that the graph will intersect the x-axis at these points:

• \( x=2 \)
• \( x=-1 \)
• \( x=-3 \)

This means the function changes sign at these locations, creating important transition points that shape the overall graph.

End Behavior of Polynomials

Understanding the end behavior of polynomial functions helps in predicting how the graph behaves as it moves towards positive or negative infinity.For the function \[ f(x) = (x-2)(x+1)(x+3) \], we recognize it as a cubic polynomial. The highest degree term, when expanded, is \[ x^3 \]. The coefficient of this term is positive, dictating the end behavior of the graph.So, how does it behave?

• As \( x \to \infty \), \( f(x) \to \infty \): The graph will rise as we move right along the x-axis.
• As \( x \to -\infty \), \( f(x) \to -\infty \): The graph will fall as we move left along the x-axis.

This tells us that the graph starts low on the left, passes through its zeros, and heads upwards on the right, maintaining the typical 'N' shape of a cubic function.

Sketching Polynomial Graphs

Once we have the zeros and understand the end behavior, we can start sketching the graph of the polynomial.First, on your coordinate plane, mark the x-intercepts, coinciding with the zeros:

• \( x = 2 \)
• \( x = -1 \)
• \( x = -3 \)

These marks will guide where the curve of the graph crosses the x-axis.Next, draw a smooth curve that reflects both the zero points and the end behavior.

• Beginning from the lower left (since \( f(x) \to -\infty \) as \( x \to -\infty \)), the curve should pass through \( x = -3 \).
• It should continue through \( x = -1 \), rising as it approaches \( x = 2 \).
• Finally, the curve stretches upwards to the right (following \( f(x) \to \infty \) when \( x \to \infty \)).

By following these steps, you can create a clear sketch of the polynomial function, capturing both the zeros and the overall direction indicated by its end behavior.