Problem 60 - 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. $$ f(x)=x^{3}-0.01 x $$

Step-by-Step Solution

Answer

The intercepts are (0,0), (0.1,0), and (-0.1,0). End behavior: \( f(x) \rightarrow -\infty \) as \( x \rightarrow -\infty \), \( f(x) \rightarrow +\infty \) as \( x \rightarrow +\infty \).

Step 1: Identify the Degree and Leading Coefficient

The function given is \( f(x) = x^3 - 0.01x \). The degree of the polynomial is the highest power of \( x \), which is 3. The leading coefficient is the coefficient of \( x^3 \), which is 1. This information about the degree and leading coefficient helps in predicting the end behavior of the graph.

Step 2: Graphing the Polynomial Function

Using a graphing calculator, plot the function \( f(x) = x^3 - 0.01x \). Observe the shape of the graph to determine the intercepts and end behavior.

Step 3: Determine the Intercepts

**Y-intercept**: Set \( x = 0 \) to find the y-intercept. \( f(0) = 0^3 - 0.01(0) = 0 \), so the y-intercept is (0, 0). **X-intercepts**: Solve the equation \( x^3 - 0.01x = 0 \) for \( x \). Factor the expression to get \( x(x^2 - 0.01) = 0 \). This gives \( x = 0 \) or \( x^2 = 0.01 \). Solving for \( x^2 = 0.01 \) yields \( x = 0.1 \) and \( x = -0.1 \). Thus, there are intercepts at (0, 0), (0.1, 0), and (-0.1, 0).

Step 4: Determine the End Behavior

For a polynomial where the degree is 3 (odd) and the leading coefficient is positive, the end behavior is: as \( x \rightarrow -\infty \), \( f(x) \rightarrow -\infty \) and as \( x \rightarrow +\infty \), \( f(x) \rightarrow +\infty \). Confirm this using the table.

Step 5: Create a Table to Confirm End Behavior

Choose large and small values of \( x \) to confirm the end behavior:- \( x = -100 \): \( f(x) = (-100)^3 - 0.01(-100) = -1000000 + 1 = -999999 \)- \( x = -10 \): \( f(x) = (-10)^3 - 0.01(-10) = -1000 + 0.1 = -999.9 \)- \( x = 10 \): \( f(x) = (10)^3 - 0.01(10) = 1000 - 0.1 = 999.9 \)- \( x = 100 \): \( f(x) = (100)^3 - 0.01(100) = 1000000 - 1 = 999999 \)These values confirm that as \( x \rightarrow -\infty \), \( f(x) \rightarrow -\infty \) and as \( x \rightarrow +\infty \), \( f(x) \rightarrow +\infty \).

Key Concepts

Degree of Polynomial

In the context of polynomial functions, the degree of a polynomial is vital in determining the overall shape and nature of the function's graph. For the function \(f(x) = x^3 - 0.01x\), the degree is 3, which is the highest power of the variable \(x\) present in the expression. Understanding the degree helps us predict some critical behaviors of the polynomial, such as the number of possible *turning points* and the function's *end behavior*.

• A polynomial of degree 3, like this one, generally has up to 3 real roots or intercepts on the x-axis.
• The graph can have up to 2 turning points since the number of turning points is always less than the degree by one.

When analyzing a polynomial, always look for the highest power term (with the biggest exponent) to identify the polynomial's degree. This dictates how the function spirals into infinity or negative infinity as \(x\) moves towards extreme values.

Leading Coefficient

The leading coefficient in a polynomial is the coefficient of the term with the highest degree. For our function \(f(x) = x^3 - 0.01x\), the leading coefficient is the number in front of \(x^3\), which is 1. The leading coefficient, along with the degree, helps predict the end behavior of the polynomial function.

• With a positive leading coefficient (like 1), a cubic polynomial's end behavior usually shows the graph rising to positive infinity as \(x\) goes to positive infinity.
• Likewise, as \(x\) tends to negative infinity, the graph will drop to negative infinity when the leading coefficient is positive.

Being familiar with these behaviors gives you insight into the general direction the function will take on either end of the x-axis, which is instrumental in graphing and understanding polynomial based models.

Intercepts

Intercepts are crucial points on a graph as they show where the function crosses the x-axis and y-axis. For \(f(x) = x^3 - 0.01x\), the intercepts are found by setting the equation to different conditions.
**Y-Intercept**:

• Set \(x = 0\). Substitute into the polynomial to get \(f(0) = 0^3 - 0.01(0) = 0\). Hence, the y-intercept is at (0, 0).

**X-Intercepts**:

• Set the polynomial equal to zero: \(x^3 - 0.01x = 0\). Factor out an \(x\) to get \(x(x^2 - 0.01) = 0\).
• From this equation, \(x = 0\), \(x = 0.1\), or \(x = -0.1\), giving us x-intercepts at (0, 0), (0.1, 0), and (-0.1, 0).

Finding these intercepts not only aids in sketching the graph accurately but is also helpful in interpreting how the graph of the function intersects with the axes.

End Behavior

The end behavior of a polynomial function describes how the graph behaves as \(x\) approaches positive or negative infinity. For the polynomial \(f(x) = x^3 - 0.01x\), the degree (odd) and leading coefficient (positive) give us a straightforward way to predict its end behavior.

• When the degree is odd and the leading coefficient is positive, as \(x \rightarrow -\infty\), \(f(x) \rightarrow -\infty\).
• Conversely, as \(x \rightarrow +\infty\), \(f(x) \rightarrow +\infty\).

This behavior is typical of positive cubic (degree 3) polynomials. For additional confirmation, calculating \(f(x)\) for large values of \(x\), such as -100 and 100, confirmed these trends by producing large negative and positive values, respectively. Recognizing end behavior is critical for understanding the full picture of the function's long-term patterns on a graph.