Problem 57 - 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. $$ f(x)=x^{4}-81 $$

Step-by-Step Solution

Answer

The x-intercepts are \( (-3, 0) \) and \( (3, 0) \), the y-intercept is \( (0, -81) \), and the end behavior shows \( f(x) \to \infty \) as \( x \to \pm \infty \).

Step 1: Identify the Type of Polynomial

The given polynomial is \( f(x) = x^4 - 81 \), which is a quartic polynomial because it is of degree 4. The leading term is \( x^4 \), which is positive.

Step 2: Graph the Polynomial

Use a graphing calculator or graphing software to plot the polynomial \( f(x) = x^4 - 81 \). Observe the shape of the graph to understand its behavior near intercepts and as \( x \to \pm \infty \).

Step 3: Find the Intercepts

To find the y-intercept, set \( x = 0 \): \( f(0) = 0^4 - 81 = -81 \). Hence, the y-intercept is at \( (0, -81) \). For x-intercepts, solve \( x^4 - 81 = 0 \). Factor it as \((x^2 - 9)(x^2 + 9) = 0\). Solve \(x^2 - 9 = 0\) to get \(x = \pm 3\). The x-intercepts are \( (3, 0) \) and \( (-3, 0) \).

Step 4: Determine the End Behavior

Look at the leading term \( x^4 \). As \( x \to \infty, f(x) \to \infty \) and \( x \to -\infty, f(x) \to \infty \), since the highest power of x is even and the coefficient is positive.

Step 5: Create a Table for End Behavior Verification

Construct a table with values of \( x \) such as \(-10, -5, 0, 5, 10\) and compute \( f(x) \) for each. The values should confirm that as \( x \to \pm \infty \), \( f(x) \to \infty \). For example, \( f(-10) = 10^4 - 81 = 9919 \) and \( f(10) = 10^4 - 81 = 9919 \), both are positive and large.

Key Concepts

Quartic Polynomials

Quartic polynomials are functions of degree 4, and they are characterized by their general form: \( f(x) = ax^4 + bx^3 + cx^2 + dx + e \), where \( a \) is non-zero. Understanding these polynomials is crucial as their graphs often have more than one turning point due to their degree. This means quartic polynomials can have up to three 'humps' or bends, unlike quadratic (degree 2) or cubic (degree 3) polynomials.

Quartic polynomials can produce a variety of shapes depending on their leading coefficients and other terms. When graphing, the focus is often on identifying these key features, such as the intercepts and end behavior, which provide significant insights into the function's graphical representation.

Graphing Polynomials

Graphing polynomial functions is a valuable skill to visually understand their behavior. A graph of a quartic polynomial can be plotted using graphing calculators or software, which shows the overall shape and features of the polynomial.

When graphing \( f(x) = x^4 - 81 \), plot several points by choosing \( x \) values like \(-10, -5, 0, 5, 10\) and calculate corresponding \( f(x) \) values. This technique provides a basic shape of the polynomial.

Graph characteristics such as intercepts and end behavior are observable, where these points highlight key attributes. Remember, the graph will generally stretch towards positive or negative y-values based on the leading coefficient's sign and the degree of the polynomial.

Intercepts of Polynomials

Intercepts are where the polynomial graph crosses the axes. They provide important insights into the function's behavior. There are two types of intercepts to identify:

• Y-intercept: To find this, set \( x = 0 \). For \( f(x) = x^4 - 81 \), substituting gives \( f(0) = -81 \). Therefore, the y-intercept is \((0, -81)\).
• X-intercepts: These occur where \( f(x) = 0 \). For our polynomial, factor \( x^4 - 81 \) into \((x^2 - 9)(x^2 + 9) = 0\). Solving gives roots at \( x = \pm 3 \), so the x-intercepts are \((3, 0)\) and \((-3, 0)\).

Intercepts help understand key turning points and transition areas where the graph switches direction.

End Behavior of Polynomials

The end behavior of a polynomial refers to the direction the graph heads as \( x \) approaches positive or negative infinity. This helps in predicting the long-term trends of the polynomial.

We explore end behavior mostly through the leading term. For \( f(x) = x^4 - 81 \), the leading term is \( x^4 \). Since \( x^4 \) is even and has a positive coefficient, both ends of the graph rise as \( x \to \pm \infty \).

To confirm predictions, we use a table with selected \( x \) values, like \(-10, -5, 0, 5, 10\). Calculating \( f(x) \) for each shows that \( f(x) \to \infty \) at these extremities, aligning with expectations. Recognizing the end behavior helps in sketching more accurate graphs.