Problem 39 - Algebra for College Students

Question

For Problems \(35-42\), (a) find the \(y\) intercepts, (b) find the \(x\) intercepts, and (c) find the intervals of \(x\) where \(f(x)>0\) and those where \(f(x)<0\). Do not sketch the graphs. $$ f(x)=x(x-6)^{2}(x+4) $$

Step-by-Step Solution

Answer

(a) y-intercept: (0,0); (b) x-intercepts: (0,0), (6,0), (-4,0); (c) f(x)>0 on (-4,0) and (6,∞), f(x)<0 on (-∞,-4) and (0,6).

Step 1: Find the y-intercept

To find the y-intercept, we set \( x = 0 \) and calculate \( f(x) \). Substituting \( x = 0 \) in the function, we have: \[ f(0) = 0 \cdot (0-6)^2 \cdot (0+4) = 0. \] Thus, the y-intercept is at the point \((0,0)\).

Step 2: Find the x-intercepts

The x-intercepts occur where \( f(x) = 0 \). We set the equation \( f(x) = x(x-6)^2(x+4) = 0 \). This product is zero when any of the factors is zero. Thus, we need to solve: \[ x = 0, \quad (x-6)^2 = 0, \quad x + 4 = 0. \] Solving these, we find \( x = 0 \), \( x = 6 \), and \( x = -4 \). Therefore, the x-intercepts are \( (0,0), (6,0), \) and \( (-4,0) \).

Step 3: Determine the sign intervals for f(x)

To find where \( f(x) > 0 \) and \( f(x) < 0 \), we analyze the sign changes over different intervals determined by the x-intercepts: \( x = -4, 0, \) and \( 6 \). Consider the intervals \((-\infty, -4)\), \((-4, 0)\), \((0, 6)\), and \((6, \infty)\).- For \( x \in (-\infty, -4) \): Test a value, e.g., \( x = -5 \). \( f(-5) = (-5)((-5)-6)^2((-5)+4) < 0 \) since there is an odd number of negative terms.- For \( x \in (-4, 0) \): Test \( x = -2 \). \( f(-2) = (-2)((-2)-6)^2((-2) + 4) > 0 \) because the negative term contributes positively when squared.- For \( x \in (0, 6) \): Test \( x = 2 \). \( f(2) = (2)(2-6)^2(2+4) < 0 \) with three positive terms and one negative.- For \( x \in (6, \infty) \): Test \( x = 7 \). \( f(7) = (7)(7-6)^2(7+4) > 0 \) with all positive terms.Thus, \( f(x) < 0 \) on \((-\infty, -4) \) and \((0, 6)\), and \( f(x) > 0 \) on \((-4, 0) \) and \((6, \infty)\).

Key Concepts

Intercepts

Intercepts are crucial points where a graph touches the axes. They help determine the behavior and position of the graph.
To find the **y-intercept**, we set the variable **x** to zero and solve for **y** or **f(x)**. In our function, \[f(x)=x(x-6)^{2}(x+4),\]when **x** is zero, the result is:\[f(0)=0(0-6)^2(0+4)=0.\]Thus, the y-intercept is \((0,0)\).Finding the **x-intercepts** involves setting the function equal to zero and solving for **x**. This means:\[x(x-6)^{2}(x+4)=0\]The function will be zero when any factor is zero, leading to:

• \(x=0\)
• \(x=6\)
• \(x=-4\)

Hence, the x-intercepts are at \((0,0), (6,0), \) and \((-4,0)\).These points indicate where the graph crosses the **x** axis.

Sign Analysis

Sign analysis helps determine in which intervals the function is positive or negative. This is key to understanding the function's behavior over the real numbers.We break down the number line into intervals based on the x-intercepts: \(-\infty, -4, 0, 6,\) and \(+\infty\). For each interval, substitute a test point to check whether the function is positive or negative.

• On \((-\infty, -4)\), choose **x = -5**. Calculating gives \[f(-5)=(-5)((-5)-6)^2((-5)+4)<0.\]
• For \((-4, 0)\), choose **x = -2**. Calculations yield\[f(-2)=(-2)((-2)-6)^2((-2)+4)>0.\]
• Between \((0, 6)\), select **x = 2**:\[f(2)=(2)(2-6)^2(2+4)<0.\]
• For the interval \((6, \infty)\), plug in **x = 7**:\[f(7)=(7)(7-6)^2(7+4)>0.\]

This analysis shows that \(f(x) < 0\) for \((-\infty, -4)\) and \((0, 6)\), but \(f(x) > 0\) on \((-4, 0)\) and \((6, \infty)\).

Factorization

Factorizing the polynomial function is breaking it down into simpler parts. This step allows us to find the x-intercepts and perform sign analysis.The function \[f(x)=x(x-6)^{2}(x+4)\]is already in its factorized form. Here, we have three factors:

• The first factor, **x**, contributes an x-intercept at \(x=0\).
• The **(x-6)^2** factor introduces a double root at \(x=6\), giving an x-intercept but influencing the sign analysis uniquely because it is squared.
• The **(x+4)** factor contributes another x-intercept at \(x=-4\).

These factors are crucial in identifying where the function changes sign and where it touches or crosses the x-axis.

Inequalities

Inequalities help identify where the value of the polynomial is greater or less than zero. They provide insight into which parts of the graph are above or below the x-axis.For the function \[f(x)=x(x-6)^{2}(x+4)\],identifying where \(f(x)>0\) and \(f(x)<0\) is essential.Using our sign analysis,

• The function is positive (+) in the intervals \((-4, 0)\) and \((6, \infty)\) where \(f(x) > 0\).
• It is negative (-) in the intervals \((-\infty, -4)\) and \((0, 6)\) where \(f(x)<0\).

Inequalities reveal not just where the graph touches but also the full range of its behavior across different domains of **x**.