Problem 28 - College Algebra with Corequisite Support

Question

For the following exercises, find the intercepts of the functions. $$ f(x)=x^{3}+27 $$

Step-by-Step Solution

Answer

The y-intercept is \((0, 27)\); the x-intercept is \((-3, 0)\).

Step 1: Understand the intercepts

In any function, the intercepts are the points where the graph of the function crosses the axes. The y-intercept occurs where \( x = 0 \), and the x-intercepts occur where \( f(x) = 0 \). For this function, \( f(x) = x^3 + 27 \), we need to find both the y-intercept and the x-intercepts.

Step 2: Find the y-intercept

The y-intercept is found by setting \( x = 0 \) in the function. \[ f(0) = 0^3 + 27 = 27 \] Thus, the y-intercept is at the point \((0, 27)\).

Step 3: Find the x-intercepts

The x-intercepts are found by setting \( f(x) = 0 \). Thus, we solve: \[ x^3 + 27 = 0 \] Rearrange it to: \[ x^3 = -27 \] Taking the cube root of both sides gives: \[ x = \sqrt[3]{-27} = -3 \]

Step 4: Identify the x-intercept

Since there is only one real cube root of -27, the x-intercept is at the point \((-3, 0)\).

Key Concepts

Understanding the Y-Intercept

The y-intercept of a function is a key concept to understand when analyzing the graph of an equation. It describes where the graph crosses the y-axis. To find the y-intercept, set the value of x to zero and solve for the function's output.
For example, in the polynomial equation \( f(x) = x^3 + 27 \), the y-intercept can be found by evaluating \( f(0) \). This yields: \[ f(0) = 0^3 + 27 = 27 \] Therefore, the point where the graph crosses the y-axis is at

• y-intercept: \( (0, 27) \)

This point gives immediate insight into the behavior of the graph, showing you where it begins on the y-axis.
Knowing the y-intercept helps in plotting the function and understanding its overall direction.

Exploring the X-Intercepts

Finding x-intercepts is essential for understanding where a graph crosses the x-axis. These are the points where the output value (f(x)) is zero. To find these for the function \( f(x) = x^3 + 27 \), we need to solve the equation \( f(x) = 0 \).
This means solving: \[ x^3 + 27 = 0 \]First, rearrange the equation to: \[ x^3 = -27 \] By taking the cube root of both sides, we find: \[ x = \sqrt[3]{-27} = -3 \]This provides us with a single x-intercept at:

• x-intercept: \( (-3, 0) \)

This solution can be verified by plugging x back into the original equation to see if it equals zero. Understanding how to find x-intercepts is crucial for sketching and analyzing polynomial graphs.

Decoding Polynomial Equations

Polynomial equations are expressions that involve variables raised to whole number powers. These equations can carry simple degrees, such as linear polynomials \( ax + b \), or more complex ones, with higher powers like cubic polynomials.
The focus equation \( f(x) = x^3 + 27 \) is a cubic polynomial with a degree of 3, signifying that the highest power of the variable x is three.
Key features of polynomial equations include:

• High-degree terms that indicate multiple solutions
• Smooth graphs without jumps
• Predictable end behavior based on the leading term

Analyzing these properties allows one to anticipate the general shape and behavior of the function's graph. Recognizing polynomial forms is central to predicting the number of intercepts and understanding the function's behavior over different intervals.

Graphing Functions Made Easy

Graphing polynomial functions is a straightforward task once you grasp the concept of intercepts and their significance. The graph of a function represents all solutions of a polynomial equation visually. With the equation \( f(x) = x^3 + 27 \), visualize the points where it crosses the axes using the x and y intercepts discussed earlier.
Here’s a step-by-step process to graph a function easily:

• Determine the y-intercept: start by plotting the point \((0,27)\)
• Find x-intercepts: plot the point \((-3,0)\) for this particular function
• Identify key features such as symmetry or direction of end behavior
• Draw a smooth curve through these points, noting that polynomial graphs have no sharp turns

Using these points as markers offers a simple guide to sketch the function's general path. Practicing these steps with various functions enhances understanding and ease in graphing more complex equations.