Problem 205 - Precalculus

Question

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$f(x)=x(14-2 x)(10-2 x)$$

Step-by-Step Solution

Answer

The intercepts are (0,0), (5,0), and (7,0). The graph rises to the left and falls to the right.

Step 1: Identify the Polynomial Function

Given the polynomial function is \[ f(x) = x(14 - 2x)(10 - 2x) \]This indicates a polynomial function of degree 3, as there are three linear factors multiplied together.

Step 2: Determine the x-intercepts

To find the x-intercepts, set each factor equal to zero:1. \(x = 0\)2. \(14 - 2x = 0 \Rightarrow x = 7\)3. \(10 - 2x = 0 \Rightarrow x = 5\)Thus, the x-intercepts of the graph are at \((0, 0), (5, 0), \text{and} (7, 0)\).

Step 3: Use a Graphing Calculator

Enter the function \(f(x) = x(14 - 2x)(10 - 2x)\) into a graphing calculator to visualize the graph and confirm the x-intercepts identified in Step 2.

Step 4: Analyze the End Behavior

For a polynomial of degree 3 with a positive leading coefficient (since the leading term is \(-4x^3\), which dominates as \(x \to \infty\) or \(x \to -\infty\)), the end behavior is:- As \(x \to -\infty\), \(f(x) \to +\infty\)- As \(x \to +\infty\), \(f(x) \to -\infty\)This behavior reflects that the graph rises to the left and falls to the right.

Key Concepts

Graphing Polynomial Functions

Graphing polynomial functions is crucial to understanding their overall behavior and characteristics. In this exercise, the polynomial function is given by \( f(x) = x(14 - 2x)(10 - 2x) \), which is a third-degree polynomial. To graph a polynomial like this, the best tool to use is a graphing calculator. Here’s how you can graph it accurately:

• First, identify the degree of the polynomial to understand its complexity. Since this function is a product of three linear factors, each factor contributes a degree, making it a degree 3 polynomial.
• The graph of a degree 3 polynomial typically has up to 2 turning points. Observing these helps in understanding the main shape of the curve.
• Enter the polynomial equation into your graphing calculator. This tool will plot the curve, allowing you to visualize its behavior at various points.
• Particularly, focus on where the graph crosses the x-axis and any turning points that appear. These features provide critical insights into the behavior of the function.

Graphing the function will reveal the interplay between its factors, indicating where it changes direction and how it approaches its x-intercepts, which is important for more complex polynomial studies.

End Behavior of Polynomials

The end behavior of a polynomial function refers to the direction in which the graph heads as the variable \(x\) approaches positive or negative infinity. Understanding the end behavior is vital because it gives you a broad sense of how the function behaves overall, outside of its immediate area around the x-intercepts.

For the function \(f(x) = x(14 - 2x)(10 - 2x)\), the end behavior is determined by the leading term. The leading term is the highest power of \(x\) when the polynomial is fully expanded, which in this case is \( -4x^3 \). This term dictates that as \(x\) changes:

• When \(x \to -\infty\), \(f(x)\) rises towards \(+\infty\). This indicates that the graph will shoot upwards to the left.
• Conversely, when \(x \to +\infty\), \(f(x)\) falls towards \(-\infty\). This suggests the graph will plummet downwards on the right.

These behaviors align with the observations of polynomial end behavior where odd-degree polynomials exhibit opposite trends at either end. It's critical to understand these behaviors, as they help in predicting how the polynomial will act as it extends beyond the graphable range.

Finding X-intercepts of Polynomials

X-intercepts of a polynomial function are the points where the function crosses the x-axis, i.e., where \(f(x)=0\). These are crucial as they portray real solutions to the equation and are pivotal for graphing and understanding polynomial behavior.

To find the x-intercepts of \(f(x) = x(14 - 2x)(10 - 2x)\), set each factor to zero and solve for \(x\).

• From the factor \(x\): setting it to zero gives the intercept \((0, 0)\).
• For the factor \(14 - 2x\): set to zero and solve for \(x\), resulting in \(x = 7\). This gives the intercept \((7, 0)\).
• For the factor \(10 - 2x\): also set this to zero and solve, leading to \(x = 5\). Hence, the x-intercept is \((5, 0)\).

The x-intercepts tell us that the graph will intersect the x-axis at these points. Knowing them aids in sketching an accurate graph of the polynomial, as they show where the function returns to a zero value before changing direction due to its turning points.