Problem 11 - A Graphical Approach to College Algebra

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 \(f\) $$f(x)=2-\frac{1}{2} x$$

Step-by-Step Solution

Answer

(a) \( f(-2) = 3 \), \( f(4) = 0 \); (b) Graph: zero at \( x = 4 \); (c) Zero: \( x = 4 \).

Step 1: Substitute and Evaluate for f(-2)

To find \( f(-2) \), substitute \( -2 \) in place of \( x \) in the function \( f(x) = 2 - \frac{1}{2}x \). Therefore, \( f(-2) = 2 - \frac{1}{2}(-2) = 2 + 1 = 3 \).

Step 2: Substitute and Evaluate for f(4)

Substitute \( 4 \) in place of \( x \) in the function \( f(x) = 2 - \frac{1}{2}x \). Therefore, \( f(4) = 2 - \frac{1}{2}(4) = 2 - 2 = 0 \).

Step 3: Graph the Linear Function

The graph of the linear function \( f(x) = 2 - \frac{1}{2}x \) is a straight line. The y-intercept is at \( (0, 2) \) and using the point from Step 2, \( (4, 0) \), draw the line. You can use these points to sketch the line on the coordinate plane.

Step 4: Analyze the Graph to Determine the Zero

The zero of the function \( f(x) \) is the x-value where the graph intersects the x-axis. From Step 3, it intersects at \( x = 4 \).

Step 5: Solve Algebraically for the Zero

To find the zero algebraically, set \( f(x) = 0 \) and solve for \( x \): \( 0 = 2 - \frac{1}{2}x \). Add \( \frac{1}{2}x \) to both sides to get \( \frac{1}{2}x = 2 \). Multiply both sides by \( 2 \) to get \( x = 4 \).

Key Concepts

Function Evaluation

Evaluating a function is all about figuring out what the output is when you plug specific values into the function. Let's break it down using our given function \( f(x) = 2 - \frac{1}{2}x \). When we want to find \( f(-2) \), we replace every \( x \) in the equation with \( -2 \).
This substitution gives us: \( f(-2) = 2 - \frac{1}{2}(-2) \). You simplify this by performing the multiplication, which results in \( 2 + 1 = 3 \).

Next, when evaluating \( f(4) \), replace \( x \) with \( 4 \), giving: \( f(4) = 2 - \frac{1}{2}(4) \). Simplify it, and you find \( f(4) = 2 - 2 = 0 \).
This step of function evaluation helps you understand how each input changes the output, which is critical in analyzing behaviors of linear functions.

• Substitute the x-value into the function
• Simplify the resulting expression
• Obtain the corresponding y-value

Graphing Linear Equations

Graphing a linear equation provides a visual representation of the function. For our linear function \( f(x) = 2 - \frac{1}{2}x \), follow these steps to draw its graph.
Start by finding the y-intercept. This happens when \( x = 0 \). Plug \( x \) as \( 0 \) in the function to get \( f(0) = 2 \). Thus, the y-intercept is \( (0, 2) \).

Next, use another known point from our function evaluation, such as \( (4, 0) \), to graph. On a coordinate plane, plot the points \( (0, 2) \) and \( (4, 0) \). These two points are sufficient, as two points determine a line.
Draw a straight line through them, and you have the graph of the linear function.

• Find the y-intercept (where x=0)
• Use another point to plot
• Draw a straight line through the points

Finding Zeros

Finding the zero of a function means finding an \( x \) value where \( f(x) = 0 \). This is also where the graph intersects the x-axis.
Following the earlier steps in graphing, we see the line intersects the x-axis at \( x = 4 \).

You can confirm this algebraically by setting \( f(x) = 0 \). Here we have \( 0 = 2 - \frac{1}{2}x \).
Solve this equation by first adding \( \frac{1}{2}x \) to both sides to get \( \frac{1}{2}x = 2 \). Multiply both sides by 2, leading to \( x = 4 \).
Finding zeros is crucial in understanding functions because it tells us the input value that results in an output of zero.

• Set the function to zero
• Solve for \( x \)
• Verify using both graph and algebra