Problem 20 - 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 \(f\) $$f(x)=\frac{x-4}{2}$$

Step-by-Step Solution

Answer

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

Step 1: Evaluating f(-2) and f(4)

To evaluate \( f(-2) \) and \( f(4) \), substitute \( x = -2 \) and \( x = 4 \) into the function \( f(x) = \frac{x - 4}{2} \). \[ f(-2) = \frac{-2 - 4}{2} = \frac{-6}{2} = -3 \] \[ f(4) = \frac{4 - 4}{2} = \frac{0}{2} = 0 \]

Step 2: Understanding the Graph of f(x)

The function \( f(x) = \frac{x - 4}{2} \) is a linear function with the standard line equation form of \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. Here, the slope \( m \) is \( \frac{1}{2} \) and the y-intercept (when \( x = 0 \)) is \( \frac{-4}{2} = -2 \). The function can be graphed as a straight line passing through these points.

Step 3: Using the Graph to Determine the Zero of f(x)

The zero of a function is where the graph intersects the x-axis. On a linear graph, this is where \( y = 0 \). The graph of \( f(x) \) is a line with a positive slope that intersects the x-axis at \( x = 4 \), which signifies the zero of the function.

Step 4: Calculating the Zero of f(x) Algebraically

To find the zero analytically, set \( f(x) \) to zero and solve for \( x \): \[ \frac{x - 4}{2} = 0 \] Multiply both sides by 2: \[ x - 4 = 0 \] Add 4 to both sides: \[ x = 4 \] Thus, the zero of \( f \) is \( x = 4 \).

Key Concepts

Evaluating Functions

Evaluating a function essentially means finding the output of the function for specific input values. It's like asking what output you get from the function when you plug in certain numbers. For instance, to evaluate the function \( f(x) = \frac{x-4}{2} \), you simply substitute the value of \( x \) with numbers given in the problem. Here, we see it with \( x = -2 \) and \( x = 4 \).

• For \( x = -2 \), plug this into the equation: \( f(-2) = \frac{-2-4}{2} = -3 \).
• For \( x = 4 \), plug this into the equation: \( f(4) = \frac{4-4}{2} = 0 \).

The results tell you what the value of the function is when \( x \) is \(-2\) and \(4\). This satisfying little bit of math is all about substitution and simplification.

Graphing Linear Functions

Graphing a linear function helps us visualize the entire set of solutions of the function as a straight line. Linear functions, like \( f(x) = \frac{x - 4}{2} \), have a slope and a y-intercept. The slope tells us how steep the line is, while the y-intercept tells us where the line crosses the y-axis.
A linear function in the form \( f(x) = mx + c \), gives us:

• Slope \( m = \frac{1}{2} \)
• Y-intercept \( c = -2 \)

To graph the function, start at the y-intercept (0, -2) and use the slope to find another point. A slope of \( \frac{1}{2} \) means that for every 1 unit increase in \( x \), \( y \) increases by 0.5 units. Draw a straight line through these points and extend it in both directions.
Graphing not only helps in visualizing the function but also plays a crucial role in analyzing it further, like finding the zeros.

Finding Zeros of Functions

The zero of a function is the input value \( x \) for which the function's output is zero. It's essentially where the graph of the function intersects the x-axis.
For the given function \( f(x) = \frac{x-4}{2} \), you can find the zero by setting the function equal to zero and solving for \( x \):

• Set \( \frac{x - 4}{2} = 0 \).
• This means that \( x - 4 = 0 \), leading to \( x = 4 \).

Graphically, this means the line crosses the x-axis at \( x = 4 \).
Zeros are important because they give us critical points which are often helpful in understanding the function's behavior and are used in calculus and algebra to solve various types of problems. By knowing the zero, you can predict other values on the function and analyze trends.