Problem 55 - College Algebra

Question

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$ f(x)=x^{3}-16 x $$

Step-by-Step Solution

Answer

X-intercepts: 0, 4, -4. Y-intercept: 0. End behavior: Falls left, rises right.

Step 1: Graph the Polynomial

To begin solving the problem, graph the polynomial function \( f(x) = x^3 - 16x \) using a graphing calculator or graphing software. Enter the function into the graphing tool and plot the graph to visualize it.

Step 2: Identify the Intercepts

Intersections with axes are called intercepts. A function intersects the x-axis when \( f(x) = 0 \). To find the x-intercepts, solve the equation \( x^3 - 16x = 0 \). This can be factored as \( x(x^2 - 16) = 0 \), resulting in \( x(x-4)(x+4) = 0 \), hence the x-intercepts are \( x = 0, 4, -4 \). The y-intercept is found by evaluating \( f(0) = 0 \), hence the y-intercept is also (0, 0).

Step 3: Determine the End Behavior

The end behavior of a polynomial function is determined by the leading term. For \( f(x) = x^3 - 16x \), the leading term is \( x^3 \). Since the degree (3) is odd and the leading coefficient (1) is positive, as \( x \to -\infty \), \( f(x) \to -\infty \) and as \( x \to \infty \), \( f(x) \to \infty \). This tells us the graph falls to the left and rises to the right.

Key Concepts

Understanding Graphing of Polynomial Functions

Graphing a polynomial function is like mapping its "blueprint." It allows us to visualize its structure and behavior. When you graph a function like \( f(x) = x^3 - 16x \), you plot points that satisfy the function’s equation.

With tools like graphing calculators or software, you simply input the function, and it will illustrate the graph for you. This graph helps to identify features such as intercepts and visualize how the polynomial behaves across its domain.

• Polynomials can curve, have turning points, and cross axes.
• The 'degree' of the polynomial, or the highest power of \( x \), often dictates the number and nature of these curves.

Identifying Intercepts in Polynomial Graphs

Intercepts are crucial parts of a polynomial graph, revealing where it interacts with the axes. For our function, \( f(x) = x^3 - 16x \), intercepts are points where the graph crosses the x-axis or y-axis.
The x-intercepts are found by setting \( f(x) = 0 \). By factoring \( x^3 - 16x = 0 \) into \( x(x-4)(x+4) = 0 \), we find x-intercepts at \( x = 0, 4, -4 \). These points show where the graph cuts across the x-axis.

The y-intercept is where the graph crosses the y-axis. This occurs when \( x = 0 \). Evaluating \( f(0) = 0 \), the y-intercept is \((0, 0)\). This means the graph passes through the origin.

Exploring End Behavior of Polynomial Functions

The end behavior of a polynomial function describes how the graph behaves as \( x \) approaches positive or negative infinity.
For \( f(x) = x^3 - 16x \), we focus on the leading term, \( x^3 \). This term dictates the end behavior since it's the highest power of \( x \).

• If the degree is odd, like in this case, one end of the graph goes one way, and the other end goes in the opposite direction.
• Given the positive leading coefficient, the graph rises to the right and falls to the left.
• This is visualized as \( x \to \infty, f(x) \to \infty \) and \( x \to -\infty, f(x) \to -\infty \).

Factoring Polynomials to Find Intercepts

Factoring is a handy tool to break down polynomials into simpler, solvable parts, especially to find intercepts.
Take \( f(x) = x^3 - 16x \). To factor, you look for common factors in each term:

• First, factor out the greatest common factor (GCF), which is \( x \). So, it factors to \( x(x^2 - 16) \).
• Recognize \( x^2 - 16 \) as a difference of squares, \( (x - 4)(x + 4) \).

This results in the fully factored form \( x(x-4)(x+4) \), making it easy to find x-intercepts at \( x = 0, 4, \) and \( -4 \).