Problem 53 - 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. $$ f(x)=x(14-2 x)(10-2 x) $$

Step-by-Step Solution

Answer

Intercepts: x-intercepts at (0,0), (7,0), (5,0); y-intercept at (0,0). End behavior: as \( x\to\infty \), \( f(x)\to\infty \); as \( x\to-\infty \), \( f(x)\to-\infty \).

Step 1: Identify the Polynomial

The given function is \( f(x) = x(14-2x)(10-2x) \). First, identify it as a polynomial. Expand the terms to identify its degree: \( f(x) = x((14)(10) - 2x(14) - 2x(10) + (2x)^2) \). Simplifying, we have \( f(x) = x(140 - 28x - 20x + 4x^2) \). Further simplification gives \( f(x) = x(140 - 48x + 4x^2) \). Once expanded, \( f(x) = 4x^3 - 48x^2 + 140x \). The polynomial is of degree 3.

Step 2: Find the Intercepts

To find the intercepts, identify where the graph crosses the axes. For the y-intercept, set \( x = 0 \): \( f(0) = 0(14-0)(10-0) = 0 \). So, the y-intercept is at (0,0). For the x-intercepts, set \( f(x) = 0 \): \(0 = x(14-2x)(10-2x). \)Solve each factor equal to zero: - \( x = 0 \) - \( 14-2x = 0 \) which gives \( x = 7 \) - \( 10-2x = 0 \) which gives \( x = 5 \). The x-intercepts are at points (0,0), (7,0), and (5,0).

Step 3: Determine the End Behavior

The end behavior of a cubic polynomial \( ax^3 + bx^2 + cx + d \) can be determined by the leading term, which is \( 4x^3 \) in this case. As \( x \to \infty \), \( 4x^3 \to \infty \), and as \( x \to -\infty \), \( 4x^3 \to -\infty \). Therefore, the end behavior is such that as \( x \to \infty \), \( f(x) \to \infty \) and as \( x \to -\infty \), \( f(x) \to -\infty \).

Step 4: Create a Table to Verify End Behavior

Select a few values of \( x \) to see the behavior of \( f(x) \) as \( x \) increases or decreases significantly. For example:- For \( x = -10 \), \( f(x) = 4(-10)^3 - 48(-10)^2 + 140(-10) = -4000 - 4800 - 1400 = -10200 \). - For \( x = 0 \), \( f(x) = 0 \).- For \( x = 10 \), \( f(x) = 4(10)^3 - 48(10)^2 + 140(10) = 4000 - 4800 + 1400 = 600 \). The table shows that as \( x \) becomes more negative, \( f(x) \) becomes more negative, and as \( x \) becomes more positive, \( f(x) \) becomes more positive, confirming the end behavior.

Key Concepts

Intercepts

Intercepts are the points where a graph intersects the x-axis and y-axis.
For a polynomial function like \( f(x) = x(14-2x)(10-2x) \), these are key features.

• X-intercepts: Found by setting the polynomial equal to zero. Here, solve \( x(14-2x)(10-2x) = 0 \). This gives x-intercepts at:
• \( x = 0 \) : The point (0,0)
• \( 14 - 2x = 0 \) resolves to \( x = 7 \) : The point (7,0)
• \( 10 - 2x = 0 \) resolves to \( x = 5 \) : The point (5,0)
• Y-intercept: Found by plugging \( x = 0 \) into the polynomial. This yields \( f(0) = 0 \), the point (0,0).

Understanding intercepts is crucial for determining the general shape and positions of a polynomial graph.

End Behavior

End behavior describes what happens to the value of \( f(x) \) as \( x \) approaches infinity or negative infinity.
In our cubic polynomial \( f(x) = 4x^3 - 48x^2 + 140x \), the end behavior is determined by the leading term, \( 4x^3 \).

• As \( x \to \infty \), \( 4x^3 \to \infty \), meaning \( f(x) \to \infty \).
• As \( x \to -\infty \), \( 4x^3 \to -\infty \), so \( f(x) \to -\infty \).

This polynomial rises to infinity as x increases and falls to negative infinity as x decreases. Thus, plot the graph extending upwards on one side and downwards on the other.

Cubic Polynomial

A cubic polynomial is a polynomial of degree three, typically of the form \( ax^3 + bx^2 + cx + d \).
The given polynomial \( f(x) = 4x^3 - 48x^2 + 140x \) is clearly cubic with a leading coefficient of 4.

• Key Characteristics of Cubic Polynomials:
• At most three real roots, according to the Fundamental Theorem of Algebra.
• Can have up to two turning points, where the derivatives are zero, indicating changes in direction.
• End behavior determined solely by the leading term \( 4x^3 \).

Understanding cubic polynomials helps in predicting their graphs, intercepts, and how they behave in different ranges.

Degree of Polynomial

The degree of a polynomial is the highest power of \( x \) that appears with a non-zero coefficient.
For \( f(x) = 4x^3 - 48x^2 + 140x \), the highest power is 3, hence a degree 3 polynomial.

• Significance:
• The degree tells us the possible number of roots and turning points.
• A degree 3 polynomial like this one can cross the x-axis three times and have up to two turning points.
• The degree also directly influences the end behavior.

Higher degree polynomials may display complex behaviors, but understanding their degree can simplify analysis, ensuring graphs are accurate and deductions sound.