Problem 29 - College Algebra with Corequisite Support

Question

For the following exercises, find the intercepts of the functions. $$ f(x)=x\left(x^{2}-2 x-8\right) $$

Step-by-Step Solution

Answer

Intercepts are at (0, 0), (4, 0), and (-2, 0).

Step 1: Expand the Function

First, expand the given function. We have the function \( f(x) = x (x^2 - 2x - 8) \). Distribute \( x \) to each term inside the parentheses: \[ f(x) = x^3 - 2x^2 - 8x \].

Step 2: Find the y-Intercept

To find the y-intercept, substitute \( x = 0 \) into the expanded function. \[ f(0) = 0^3 - 2(0)^2 - 8(0) = 0 \]. Thus, the y-intercept is at the point \( (0, 0) \).

Step 3: Find the x-Intercepts

To find the x-intercepts, set \( f(x) = 0 \) and solve for \( x \). The equation is \( x^3 - 2x^2 - 8x = 0 \). Factor out \( x \): \[ x(x^2 - 2x - 8) = 0 \]. This gives us one solution \( x = 0 \).

Step 4: Solve the Quadratic Factor

Now solve the quadratic equation \( x^2 - 2x - 8 = 0 \). Factor this into \((x - 4)(x + 2) = 0\). Thus, the solutions are \( x = 4 \) and \( x = -2 \).

Step 5: List All Intercepts

The intercepts of the function are all the solutions where the graph intersects the axes: the y-intercept is \( (0, 0) \), and the x-intercepts are \( (0, 0), (4, 0), (-2, 0) \).

Key Concepts

Factoring Polynomials

Factoring polynomials is a vital skill in algebra, particularly when dealing with functions and their intercepts. Essentially, factoring involves breaking down a polynomial into simpler terms or factors that, when multiplied, give back the original polynomial.

• It helps simplify equations, making it easier to solve for variable values.
• Factorization reveals vital properties of the polynomial, such as roots and intercepts.

For the exercise provided, the polynomial is already factored partially: \( f(x) = x(x^2 - 2x - 8) \). The first term, \( x \), separates immediately, indicating that \( x = 0 \) is a solution. The second term, \( x^2 - 2x - 8 \), requires further factoring to identify all solutions.
By inspecting the quadratic \( x^2 - 2x - 8 \), we identify two numbers that multiply to -8 and add to -2. These numbers are -4 and +2, allowing the expression to factor further into \((x - 4)(x + 2)\). Through factoring, we discover that the polynomial can be expressed as a product of simpler terms, making it easier to find x-intercepts.

X-intercept

The x-intercept of a function is where the graph crosses the x-axis. At this point, the y-value is zero, and only the x-value matters.

• It involves setting the entire function equal to zero and solving for \( x \).
• The solutions represent all x-values where the function touches or crosses the x-axis.

In the step-by-step solution, after factoring \( f(x) = x(x^2 - 2x - 8) \), we set it equal to zero: \( f(x) = 0 \). The solutions \( x = 0 \), \( x = 4 \), and \( x = -2 \) provide the points \( (0, 0) \), \( (4, 0) \), and \( (-2, 0) \). Each represents a location on the x-axis where the polynomial's graph intersects it.The x-intercepts are crucial in understanding the graph's behavior and where it changes direction as it crosses the axis.

Y-intercept

The y-intercept of a function is the coordinate where the graph crosses the y-axis. It's determined by setting \( x \) to zero in the function and solving for \( y \).

• Typically, this involves a simple calculation since it isolates terms that include x, leaving a constant that reveals the y-value.
• The y-intercept represents the initial value or starting point of the function when x is zero.

In our example, substituting \( x = 0 \) into the function gives: \[ f(0) = 0^3 - 2(0)^2 - 8(0) = 0 \]. This computation reveals that the y-intercept is \( (0, 0) \). The y-intercept provides a quick insight into where the graph starts from a vertical perspective and illustrates symmetry in some polynomial graphs that pass through the origin, like in this exercis.

Quadratic Equation

A quadratic equation is a type of polynomial equation of the form \( ax^2 + bx + c = 0 \). It features a degree of two, which indicates it will generally produce two solutions or roots.

• The solutions of a quadratic equation are found through factoring, using the quadratic formula, or via completing the square.
• In terms of graphing, these solutions correspond to the points where the function crosses the x-axis, known as x-intercepts.

In the given exercise, the quadratic \( x^2 - 2x - 8 \) is identified and factored as \( (x - 4)(x + 2) = 0 \). Solving this yields \( x = 4 \) and \( x = -2 \), which are the x-values where the graph intersects with the x-axis. Understanding how to work with quadratics equips students with tools to analyze these crossing points effectively, thus unlocking more about the function's behavior and structure.