Problem 11 - A Graphical Approach to Precalculus with Limits

Question

Work each problem related to linear functions. (a) Evaluate \(f(-2)\) and \(f(4)\) (b) Graph \(f\). How can the graph of \(f\) be used to determine the zero of \(f ?\) (c) Find the zero of \(f\) $$f(x)=-\frac{1}{2} x+2$$

Step-by-Step Solution

Answer

Evaluate: \(f(-2) = 3\), \(f(4) = 0\). Zero: \(x = 4\).

Step 1: Evaluate f(-2)

Substitute -2 into the function equation:\[ f(-2) = -\frac{1}{2}(-2) + 2 \]First calculate \(-\frac{1}{2} \times (-2)\), which is 1. Then, add 2:\[ f(-2) = 1 + 2 = 3 \]

Step 2: Evaluate f(4)

Substitute 4 into the function equation:\[ f(4) = -\frac{1}{2}(4) + 2 \]Calculate \(-\frac{1}{2} \times 4\), which is -2. Then, add 2:\[ f(4) = -2 + 2 = 0 \]

Step 3: Graph f

To graph the function, use the function equation \(f(x) = -\frac{1}{2}x + 2\). The y-intercept is 2 (where \(f(0) = 2\)), and the slope is \(-\frac{1}{2}\). This indicates a line starting at (0, 2) and decreasing by 1 unit down for every 2 units it moves to the right. Plot points such as (0, 2) and (4, 0) to draw the graph.

Step 4: Use the Graph to Determine the Zero

The zero of the function is the point where it crosses the x-axis. From the graph, determine where \(f(x) = 0\). Since the point (4, 0) is on the graph, the zero is x = 4.

Step 5: Find the Zero Algebraically

To find the zero of the function, solve for \(x\) in the equation:\[ -\frac{1}{2}x + 2 = 0 \]Move 2 to the other side:\[ -\frac{1}{2}x = -2 \]Multiply both sides by -2:\[ x = 4 \]Thus, the zero of the function is \(x = 4\).

Key Concepts

Function Evaluation

Function evaluation is the process of finding the value of a function for a specific input value. Essentially, you're determining what the output will be when you "plug in" a number for the variable in your function. Let's see how it works:

• Given a function, such as \( f(x) = -\frac{1}{2}x + 2 \), evaluate it by substituting the value of \( x \) with the given number.
• For example, to evaluate \( f(-2) \), replace \( x \) with \(-2\): \,\( f(-2) = -\frac{1}{2}(-2) + 2 \). Calculate this step-by-step, first finding \(-\frac{1}{2} \times (-2)\), which equals 1, and then adding 2 to get the final result of 3.
• Similarly, evaluate \( f(4) \) by substituting \( x \) with 4: \,\( f(4) = -\frac{1}{2}(4) + 2 \). Perform the multiplication \(-\frac{1}{2} \times 4\), resulting in -2, and add 2 to arrive at 0.

Function evaluation is vital for understanding the output behavior of linear functions at specific points, which helps in deeper analysis such as graphing.

Graphing Linear Functions

Graphing linear functions involves representing the function as a line on the coordinate plane. This helps visualize how the function behaves. Here's a straightforward guide:

• A linear function like \( f(x) = -\frac{1}{2}x + 2 \) can be graphed using its slope and y-intercept.
• The y-intercept is where the line crosses the y-axis, which occurs when \( x = 0 \). Here, the y-intercept is 2, so you start plotting the point (0, 2).
• The slope \,\( -\frac{1}{2} \) indicates how steep the line is and the direction it goes. It tells us to move down 1 unit in y for every 2 units we move right in x. From (0, 2), a point such as (4, 0) can be drawn.

Connect these points with a straight line. You'll see a downward trend due to the negative slope. Graphing allows you to easily identify important characteristics, including the function's zeros.

Finding Zeros of Functions

Zeros of a function, often called "roots," are the points where the function's value is zero. These points lie on the x-axis. Here's how you find them:

• Begin with setting the function equal to zero. For \( f(x) = -\frac{1}{2}x + 2 \), solve \( -\frac{1}{2}x + 2 = 0 \).
• Next, move the constant term to the other side: \( -\frac{1}{2}x = -2 \).
• To isolate \( x \), multiply both sides by \(-2\) to get \( x = 4 \).

This process shows that the zero of the function occurs at \( x = 4 \). You can verify this zero by observing where the graph of \( f \) intersects the x-axis, confirming \( x = 4 \) as the zero. Understanding and finding zeros are fundamental in solving equations and analyzing the function's behavior.