Problem 30 - College Algebra with Corequisite Support

Question

For the following exercises, find the intercepts of the functions. $$ f(x)=(x+3)\left(4 x^{2}-1\right) $$

Step-by-Step Solution

Answer

The intercepts are y-intercept (0, -3) and x-intercepts (-3, 0), (1/2, 0), (-1/2, 0).

Step 1: Identify Intercepts

An intercept is a point where the graph crosses the x-axis or y-axis. For a function \( f(x) \), the x-intercepts are solutions to \( f(x) = 0 \), and the y-intercept is found by evaluating \( f(0) \).

Step 2: Determine the y-intercept

To find the y-intercept, substitute \( x = 0 \) in the function: \( f(x) = (x+3)(4x^2-1) \). This yields \( f(0) = (0 + 3)(4(0)^2 - 1) = 3(-1) = -3 \). Thus, the y-intercept is \((0, -3)\).

Step 3: Find x-intercepts by setting f(x) equal to 0

Set \( f(x) = 0 \): \((x+3)(4x^2-1) = 0\). Use the zero product property, which states that if a product is zero, at least one of the factors must be zero.

Step 4: Solve for x in \(x + 3 = 0\)

Set \( x + 3 = 0 \) and solve for \( x \). This gives \( x = -3 \). So, one x-intercept is \((-3, 0)\).

Step 5: Solve for x in \(4x^2 - 1 = 0\)

Set \( 4x^2 - 1 = 0 \), then solve for \( x \): \[ 4x^2 = 1 \]\[ x^2 = \frac{1}{4} \]Take the square root of both sides: \( x = \pm \frac{1}{2} \). Thus, the x-intercepts are \( \left( \frac{1}{2}, 0 \right) \) and \( \left( -\frac{1}{2}, 0 \right) \).

Key Concepts

Understanding x-intercepts

To understand x-intercepts or roots of a polynomial function, it's important to recognize that these are the points where the graph of the function touches or crosses the x-axis. At these points, the value of the function, represented as \( f(x) \), is zero. Essentially, solving for x-intercepts means finding the values of \( x \) where \( f(x) = 0 \).

For the function \( f(x) = (x+3)(4x^2-1) \), you set it equal to zero and solve for \( x \):

• First, find when \( x+3 = 0 \). Solving gives \( x = -3 \). This means there is an x-intercept at \( (-3, 0) \).
• Next, consider \( 4x^2 - 1 = 0 \). Working through the equation solves to \( x = \pm \frac{1}{2} \). This results in x-intercepts at \( \left( \frac{1}{2}, 0 \right) \) and \( \left( -\frac{1}{2}, 0 \right) \).

All these x-intercepts are crucial for graphing the function and understanding its behavior along the x-axis.

Determining the y-intercept

Finding the y-intercept of a function answers the question: "At what point does the graph cross the y-axis?" This occurrence is where the value of \( x \) is zero, and you're simply calculating \( f(0) \).

In our example with \( f(x) = (x+3)(4x^2-1) \), substituting \( x = 0 \) into the equation will help us find the y-intercept:

• Compute \( f(0) = (0 + 3)(4(0)^2 - 1) = 3(-1) = -3 \).

Thus, the y-intercept is at the point \( (0, -3) \). This means the graph will intersect the y-axis at this point. Knowing the y-intercept provides a starting location when sketching the graph of a function.

Applying the Zero Product Property

The zero product property is a helpful mathematical rule stating that if a product of multiple factors equals zero, then at least one of the factors must be zero.

This property allows us to break down more complex polynomial equations into simpler parts, which can then be individually solved. Here's how it applies to our polynomial function \( f(x) = (x+3)(4x^2-1) \):

• Setting the entire expression equal to zero gives \((x+3)(4x^2-1) = 0\).
• Using the zero product property, either \(x+3 = 0\) or \(4x^2 - 1 = 0\).

By solving each equation separately, we find the x-intercepts.

The zero product property is vital in simplifying and solving polynomial equations efficiently. It enables you to identify intercepts of a function graphically represented, offering insights into where the graph will meet the x-axis.