Problem 52 - College Algebra

Question

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

Step-by-Step Solution

Answer

X-intercepts: \(x = 0, 3, -3\); Y-intercept: \((0, 0)\); End behavior: \(f(x) \to \infty\) as \(x \to \infty\), \(f(x) \to -\infty\) as \(x \to -\infty\).

Step 1: Input Function into Calculator

Begin by inputting the polynomial function \( f(x) = x(x-3)(x+3) \) into the graphing calculator. To do this, access the function or graphing mode of your calculator, and type the given expression directly into the function slot.

Step 2: Graph the Function

Once the function has been entered correctly, initiate the graphing feature of the calculator. Observe the graph that is displayed. Examine the shape and position of the curve on the coordinate plane.

Step 3: Determine the X-Intercepts

The x-intercepts occur where the graph crosses the x-axis, meaning \( f(x) = 0 \). For the given function, set each factor of \( f(x) = x(x-3)(x+3) \) to zero: \( x = 0 \), \( x-3 = 0 \), and \( x+3 = 0 \). Thus, the x-intercepts are at \( x = 0 \), \( x = 3 \), and \( x = -3 \). Verify these points on the graph.

Step 4: Identify the Y-Intercept

The y-intercept is the point where the graph crosses the y-axis, which occurs when \( x = 0 \). Substitute \( x = 0 \) into the original function: \( f(0) = 0(0-3)(0+3) = 0 \). Therefore, the y-intercept is at \( (0, 0) \). Confirm this point on the graph.

Step 5: Analyze the End Behavior

Examine how the graph behaves as \( x \) approaches positive and negative infinity. Since \( f(x) = x^3 - 9x \) is a cubic polynomial with a positive leading coefficient \( 1 \), the end behavior is such that as \( x \to \infty \), \( f(x) \to \infty \) and as \( x \to -\infty \), \( f(x) \to -\infty \). Confirm this trend in the calculator's graph.

Key Concepts

X-Intercepts

Finding the x-intercepts of a polynomial is crucial for understanding where the graph of the function crosses the x-axis. To determine the x-intercepts, you need to set the polynomial function equal to zero and solve for the variable. For the function given, \( f(x) = x(x-3)(x+3) \), this means you solve for when each factor is zero:

• \( x = 0 \)
• \( x-3 = 0 \) which simplifies to \( x = 3 \)
• \( x+3 = 0 \) which simplifies to \( x = -3 \)

So, the x-intercepts are at the points \( (0,0) \), \( (3,0) \), and \( (-3,0) \). This insight not only helps in sketching the graph of the function but also in understanding the roots of the polynomial. Make sure to verify these points using a graphing calculator to see if the curve crosses the axes at these values.

Y-Intercept

The y-intercept of a polynomial function is where the graph crosses the y-axis. This occurs at the point where \( x = 0 \). Calculating the y-intercept is straightforward and involves substituting \( x = 0 \) into the function. For the function \( f(x) = x(x-3)(x+3) \), the calculation would look like this:\[f(0) = 0(0-3)(0+3)\]which simplifies to \( f(0) = 0 \). Therefore, the y-intercept is at the point \( (0,0) \). This point is both an x-intercept and a y-intercept, which makes it a crucial feature of the graph. Looking at the graph created by your calculator can confirm this point, as the curve should intersect the y-axis exactly at this spot. Understanding the y-intercept is essential for grasping how the function translates in the coordinate plane.

End Behavior

The end behavior of a polynomial function describes what happens to the values of \( f(x) \) as \( x \) approaches positive or negative infinity. This behavior is determined by the degree and the leading coefficient of the polynomial. For the function \( f(x) = x(x-3)(x+3) \), which expands to \( f(x) = x^3 - 9x \), the leading term is \( x^3 \). This tells you that the graph of the function will end in certain directions:

• As \( x \to \infty \) (i.e., as \( x \) moves to the far right), \( f(x) \to \infty \) indicating the graph rises to infinity.
• As \( x \to -\infty \) (i.e., as \( x \) moves to the far left), \( f(x) \to -\infty \) showing that the graph falls to negative infinity.

Understanding the end behavior helps paint a full picture of the graph's shape. Visualizing this with a graphing calculator can offer an immediate confirmation and aid in comprehending how polynomials behave as their variables grow large.

Graphing Calculators

Graphing calculators are powerful tools for visualizing polynomial functions. By inputting the function and observing its graph, you gain insight into the behavior and key characteristics of the function.When using a graphing calculator, the first step is to input the polynomial, such as \( f(x) = x(x-3)(x+3) \), into the appropriate function slot and then select the graphing mode to visualize it. Benefits of a graphing calculator:

• They provide a visual image of the function's graph.
• You can clearly see the x-intercepts and y-intercepts.
• You can examine the end behavior of the function.
• They help to quickly verify calculated points on the curve.

Exploring these features directly on the graph enhances your understanding of the polynomial’s properties, making graphing calculators a valuable resource for any math student. This technological aid supports learning by providing immediate feedback and aiding in the discovery of mathematical concepts visually.