Problem 24 - Algebra for College Students

Question

Find the \(x\) -intercepts and discuss the behavior of the graph of each polynomial function at its \(x\) -intercepts. See Example 2 . $$f(x)=3 x-2$$

Step-by-Step Solution

Answer

The x-intercept is \( x = \frac{2}{3} \). The graph crosses the x-axis at this point.

Step 1 - Set the Function to Zero

To find the x-intercepts, set the polynomial function equal to zero: \[ f(x) = 0 \]

Step 2 - Solve for x

The given function is \( f(x) = 3x - 2 \). Set it to zero and solve for x: \[ 3x - 2 = 0 \]

Step 3 - Isolate x

Add 2 to both sides of the equation to isolate the term with x: \[ 3x = 2 \]

Step 4 - Divide by the Coefficient of x

Divide both sides by 3 to solve for x: \[ x = \frac{2}{3} \]

Step 5 - Discuss the Behavior at the x-intercept

The polynomial \( f(x) = 3x - 2 \) is a linear function with a single x-intercept at \( x = \frac{2}{3} \). The behavior of the graph at this intercept is that it crosses the x-axis, indicating a straightforward, linear behavior.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that can represent various types of curves on a graph. A basic example is a linear polynomial, which takes the form of f(x) = ax + b where a and b are constants. In this equation, the highest power of the variable x is 1, making it a first-degree polynomial. Linear polynomials form straight lines when graphed. Higher-degree polynomials include quadratic functions (second-degree), cubic functions (third-degree), and beyond. Each type offers its unique graph shapes and behaviors.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. Let's consider the function given in the exercise: f(x) = 3x - 2. To find the x-intercept, we set the function equal to zero and solve for x.
Step-by-step:

• Start by setting the function equal to zero. In this case, we get: f(x) = 0 or 3x - 2 = 0
• Next, isolate the term with x. Here we add 2 to both sides of the equation: 3x = 2
• Finally, solve for x by dividing both sides of the equation by 3: x = 2/3

The solution is x = 2/3. This means the graph of f(x) = 3x - 2 intersects the x-axis at x = 2/3.

Graph Behavior at Intercepts

When studying polynomial functions, particularly their intercepts, it's important to understand how the graph behaves at those points. In a linear function like f(x) = 3x - 2, the x-intercept is where the graph crosses the x-axis. For this function, the x-intercept is at x = 2/3. Here, the graph will intersect the x-axis at this point and continue in a straight line due to its linear nature. Higher-degree polynomials have different behaviors. For example:

• Quadratic functions (second-degree polynomials) can touch or cross the x-axis at their intercepts.
• Cubic functions (third-degree polynomials) often exhibit 'S' shaped curves, crossing the x-axis at their intercepts.

Understanding this behavior helps in predicting and sketching the polynomial functions accurately.