Problem 54 - College Algebra

Question

Graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior. $$f(x)=x(14-2 x)(10-2 x)^{2}$$

Step-by-Step Solution

Answer

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

Step 1: Enter the Function into Calculator

Enter the polynomial function \( f(x) = x(14-2x)(10-2x)^2 \) into your graphing calculator. Make sure you input it correctly to match the expression, including all operations and exponents.

Step 2: Set a Suitable Viewing Window

Adjust the viewing window of your calculator to ensure you can see all relevant features of the graph. This might involve setting the x-range and y-range to values where the graph is clearly visible.

Step 3: Identify the x-Intercepts

Find the points where the graph crosses the x-axis. For \( f(x) = x(14-2x)(10-2x)^2 \), set \( f(x) = 0 \) and solve for \( x \). These solutions include \( x = 0 \), \( x = 7 \), and \( x = 5 \). These values should match the graph's x-intercepts.

Step 4: Determine the y-Intercept

The y-intercept occurs when \( x = 0 \). Substitute \( x = 0 \) into the function to find \( f(0) = 0(14-2\times0)(10-2\times0)^2 = 0 \). So, the y-intercept is \( (0,0) \).

Step 5: Analyze the End Behavior

Examine the behavior of the graph as \( x \) approaches positive and negative infinity. Since \( f(x) = x(14-2x)(10-2x)^2 \) is a third-degree polynomial (degree from the factor \((10-2x)^2\) contributes \( 2 \)), the end behavior is determined by the leading term \(-4x^3\). As \( x \to -\infty \), \( f(x) \to \infty \) and as \( x \to \infty \), \( f(x) \to -\infty \).

Key Concepts

graphing calculator

A graphing calculator is a powerful tool that can help us visualize complex equations and functions. It's especially handy when dealing with polynomial functions like the one in our exercise. By plotting the graph of a function, we can easily determine key features such as intercepts and overall behavior. To graph a function with a calculator:

• First, input the entire expression correctly. For the function \( f(x) = x(14-2x)(10-2x)^2 \), ensure each term, multiplication, and power is entered precisely.
• Adjust the viewing window. This often involves setting appropriate ranges for both \( x \) and \( y \)-axes so that you can see the relevant parts of the graph. If the graph appears too small or some intersects aren't visible, tweak these settings accordingly.

Using graphing calculators, therefore, bridges the gap between theory and visual understanding, making it simpler to interpret polynomial functions.

x-intercepts

X-intercepts are the points where a function's graph crosses the x-axis. For polynomial functions, these are the real solutions where \( f(x) = 0 \). In mathematical terms, it means finding the roots of the polynomial.

• For the given function \( f(x) = x(14-2x)(10-2x)^2 \), find each factor that equals zero.
• Setting each factor to zero gives the solutions: \( x=0 \), \( x=7 \), and \( x=5 \). These correspond to the x-intercepts.
• Checking with a graphing calculator helps confirm these intercepts visually on the plotted graph.

Identifying x-intercepts is crucial as they provide insights into where a graph starts crossing or intersecting the x-axis.

y-intercepts

The y-intercept of a function is the point where the graph crosses the y-axis, which happens when \( x=0 \). Finding it involves simple substitution into the function.

• For \( f(x) = x(14-2x)(10-2x)^2 \), set \( x = 0 \) and solve for \( f(x) \).
• Calculate as follows: \( f(0) = 0(14-2\times0)(10-2\times0)^2 \), which results in \( f(0)=0 \).
• This tells us the y-intercept is \( (0,0) \).

The y-intercept gives a quick reference point on the graph, making it easier to understand the graph's starting position.

end behavior

End behavior describes how a function behaves as \( x \) moves towards positive or negative infinity. The degree and leading coefficient of a polynomial help determine this. For a third-degree polynomial like \( f(x) = x(14-2x)(10-2x)^2 \), these aspects are especially informative:

• The function’s leading term, found by multiplying the highest degree terms in each factor, is \( -4x^3 \).
• As \( x \rightarrow -\infty \), the function \( f(x) \rightarrow \infty \).
• As \( x \rightarrow \infty \), the function \( f(x) \rightarrow -\infty \).

Understanding end behavior helps predict the general direction of the graph over very large or very small \( x \) values.

third-degree polynomial

A third-degree polynomial, also known as a cubic polynomial, is an expression where the highest power of \( x \) is 3. It typically features a wiggle or wave-like graph owing to its points of inflection and turning points.

• The general form is \( ax^3 + bx^2 + cx + d \), though it can be expressed as a product of its factors.
• In our exercise, \( f(x) = x(14-2x)(10-2x)^2 \) expands to include a degree 3, determined by its terms when fully distributed.
• These polynomials can have up to three real roots and can change direction twice, creating a curve.

Analyzing third-degree polynomials involves recognizing these characteristics, aiding in graph sketching and real-world applications.