Problem 58 - College Algebra

Question

Graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$f(x)=-x^{3}+x^{2}+2 x$$

Step-by-Step Solution

Answer

X-intercepts and y-intercept are at (0,0), (1,0), and (-2,0); ends go down right and up left.

Step 1: Enter the Function

Use your calculator to enter the polynomial function. Access the graphing feature and type in the function: \(f(x) = -x^3 + x^2 + 2x\). Make sure the equation is entered correctly.

Step 2: Graph the Function

After entering the function, press the graph button to display the graph of the function. Observe how the curve is shaped by noting the turning points and direction of the curve at the extremes.

Step 3: Identify the Intercepts

Examine the graph to find where it crosses the axes. The x-intercepts are the points where the curve crosses the x-axis (these will be the roots of the function). The y-intercept is the point where the curve crosses the y-axis.

Step 4: Determine the X-intercepts

Look for the points on the x-axis where the graph crosses. Using the trace or calculate zero feature on the calculator, find approximate values for the x-intercepts.

Step 5: Determine the Y-intercept

The y-intercept is the value of the function when \(x = 0\). Plug \(x = 0\) into the function: \(f(0) = 0\), therefore, the y-intercept is at (0,0).

Step 6: Analyze End Behavior

Consider the degree and leading coefficient of the polynomial. Since the function is \(-x^3 + x^2 + 2x\), it has a negative leading coefficient and an odd degree. This means as \(x\) approaches infinity, \(f(x)\) will approach negative infinity. As \(x\) approaches negative infinity, \(f(x)\) will approach positive infinity.

Key Concepts

Using a Graphing Calculator for Polynomial Functions

A graphing calculator is a powerful tool when working with polynomial functions. It allows you to easily visualize complex functions and analyze their properties. To graph a polynomial like \(f(x) = -x^3 + x^2 + 2x\), follow these steps:

• Enter the equation into your calculator's graphing feature precisely to avoid mistakes.
• Once entered, use the graph button to display the visual representation of the function.

This visual representation can help you understand the function's behavior, its intercepts, and other important features by observing the curve the graph creates.

Identifying X-intercepts

The x-intercepts of a polynomial function are the values of \(x\) where the graph crosses or touches the x-axis. These are also known as the roots or solutions of the equation.To find the x-intercepts using a graphing calculator:

• Watch for points where the curve crosses the x-axis.
• Use the 'Trace' or 'Zero' feature on your calculator to identify the exact x-values of these points.

For example, in the function \(f(x) = -x^3 + x^2 + 2x\), the use of these features will help you find approximate values of the x-intercepts. Knowing these points is crucial for understanding where the function changes sign, as well as its behavior around those points.

Identifying Y-intercepts

The y-intercept is a point where the graph of the function crosses the y-axis. This occurs when \(x=0\). It is a crucial value as it provides information on the function's vertical position when \(x\) is zero.To find the y-intercept:

• Substitute \(x=0\) into the polynomial function.
• For \(-x^3 + x^2 + 2x\), compute \(f(0) = 0\).

Therefore, the y-intercept for this function is at the point (0, 0). This method gives a straightforward way to locate where the graph intersects the y-axis, which is crucial for plotting the graph accurately on paper or digital media.

Understanding End Behavior

The end behavior of a polynomial function describes how the function behaves as \(x\) becomes very large or very small. This can be understood by analyzing the degree and leading coefficient.For the function \(-x^3 + x^2 + 2x\):

• Its degree (3) is odd, and the leading coefficient (-1) is negative.
• For odd-degree polynomials with negative leading coefficients, as \(x\) approaches infinity, \(f(x)\) will head toward negative infinity.
• Conversely, as \(x\) approaches negative infinity, \(f(x)\) will rise towards positive infinity.

Understanding these patterns allows you to predict the general direction of the graph's ends without having to plot every point. This is important for sketching graphs and analyzing polynomial trends efficiently.