Problem 59 - College Algebra with Corequisite Support

Question

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.For the following exercises, make a table to confirm the end behavior of the function. $$ f(x)=x^{3}-2 x^{2}-15 x $$

Step-by-Step Solution

Answer

X-intercepts: 0, -3, 5. End behavior: rises right, falls left. Y-intercept: 0.

Step 1: Understanding the Function

We are given the polynomial function \(f(x) = x^3 - 2x^2 - 15x\). This is a cubic function, meaning it is of degree 3. We will analyze its graph to find its intercepts and end behavior.

Step 2: Using a Graphing Calculator

Enter the function \(f(x) = x^3 - 2x^2 - 15x\) into a graphing calculator. Observe the graph to identify where the function crosses the x-axis (x-intercepts) and y-axis (y-intercept).

Step 3: Determining the Intercepts

By examining the graph, note the x-intercepts where the curve crosses the x-axis. These occur at points where \(f(x) = 0\). The y-intercept is where the graph crosses the y-axis, which is the constant term in the function (in this case 0, since there is no constant term other than 0).

Step 4: Analyzing the End Behavior

The end behavior of a polynomial depends on the leading term, which here is \(x^3\). For large positive or negative values of \(x\), the \(x^3\) term dominates, causing the graph to rise to positive infinity as \(x\) goes to positive infinity, and fall to negative infinity as \(x\) goes to negative infinity.

Step 5: Confirming End Behavior with Calculations

Create a table with selected values of \(x\) such as -10, -1, 0, 1, and 10 to see the trend in \(f(x) = x^3 - 2x^2 - 15x\). Evaluate \(f(x)\) for these values to observe how the function behaves for large magnitudes of \(x\). This confirms: - As \(x\rightarrow + ext{infinity}, f(x)\rightarrow + ext{infinity}\).- As \(x\rightarrow - ext{infinity}, f(x)\rightarrow - ext{infinity}\).

Step 6: Conclusion

Based on the graph and created table, the x-intercepts are approximately at \(x = 0, x = -3,\) and \(x = 5\). The y-intercept is at \(y = 0\). The end behavior is confirmed as described.

Key Concepts

Cubic Functions

A cubic function is a type of polynomial that is characterized by the highest exponent of its variable being three. This means it can be expressed in the form of: \( f(x) = ax^3 + bx^2 + cx + d \). Here, \(a, b, c,\) and \(d\) are constants, and \(a\) cannot be zero.
Cubic functions often have interesting properties:

• They have an "S" shaped curve, which is not seen in linear or quadratic functions.
• The graph can cross the x-axis at most three times.
• Unlike quadratic functions, cubic functions do not have a maximum or minimum that constrain the entire graph, as their ends extend to both positive and negative infinity.

Understanding these features helps in sketching cubic functions by hand and analyzing their behavior, especially the turning points and intercepts.

Graphing Polynomial Functions

Graphing polynomial functions can reveal a lot about their characteristics and behavior. When you graph a function like \( f(x) = x^3 - 2x^2 - 15x \), you should look at several aspects delivered by the visual representation:

• Shape and Turning Points: For cubic functions, the graph usually has a swooping curve with one or two turning points, depending on the specific coefficients. The turning points are where the direction of the curve changes.
• Slope: Observe where the function is increasing or decreasing on different intervals of the x-axis. The slope indicates the speed of change for the function.

Using a graphing calculator can be very helpful since it quickly shows these characteristics. It is also a key tool in confirming and exploring the behavior of more complex polynomial functions.

Intercepts in Graphs

Intercepts are key points that help in understanding the main features of polynomial graphs:

• X-intercepts: These are points where the graph crosses the x-axis. For our function \( f(x) = x^3 - 2x^2 - 15x \), this occurs when \( f(x) = 0 \). Solutions to this equation will give the x-intercepts, which in this case are \(x = 0, x = -3,\) and \(x = 5\).
• Y-intercept: The graph's y-intercept is where it crosses the y-axis. For a polynomial, this is simply the value of the function when \(x = 0\). For the given function, substituting \(x = 0\) shows that the y-intercept is at \(y = 0\).

Knowing the intercepts can give a great starting point for sketching the graph and understanding the overall behavior of the function.

End Behavior of Functions

The end behavior of functions describes how the function behaves as \(x\) approaches positive and negative infinity. For polynomial functions, the leading term greatly affects this characteristic:
For the function \( f(x) = x^3 - 2x^2 - 15x \), the leading term is \(x^3\). This term decides that:

• As \( x \to +\infty \), \( f(x) \to +\infty \)
• As \( x \to -\infty \), \( f(x) \to -\infty \)

This is because the positive coefficient of \(x^3\) causes the function to rise to positive infinity as \( x \) becomes very large, and fall to negative infinity when \( x \) is very negative. Watching these trends allows you to predict the far-off behavior of the function, which is often crucial for understanding the graph's full picture.