Problem 13 - 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 \(f\) $$f(x)=\frac{1}{3} x$$

Step-by-Step Solution

Answer

\( f(-2) = -\frac{2}{3} \), \( f(4) = \frac{4}{3} \); zero of \( f \) is \( x = 0 \).

Step 1: Evaluate f(x) at x = -2

To find \( f(-2) \), substitute \( -2 \) for \( x \) in the function: \[ f(x) = \frac{1}{3} x \] so, \[ f(-2) = \frac{1}{3}(-2) = -\frac{2}{3}. \]

Step 2: Evaluate f(x) at x = 4

Substitute \( 4 \) for \( x \) in the function \( f(x) = \frac{1}{3} x \): \[ f(4) = \frac{1}{3}(4) = \frac{4}{3}. \]

Step 3: Graph f(x)

The function \( f(x) = \frac{1}{3} x \) is a linear function. Its graph is a straight line that passes through the origin. The slope of the line is \( \frac{1}{3} \), which means for each unit increase in \( x \), \( f(x) \) increases by \( \frac{1}{3} \).

Step 4: Determine the zero of f(x) from the graph

The zero of \( f \) is the \( x \)-value where \( f(x) = 0 \). From the graph of \( f(x) = \frac{1}{3} x \), it is evident that the line passes through the origin, hence the zero is at \( x = 0 \).

Step 5: Find the zero algebraically

Set \( f(x) \) equal to zero and solve for \( x \): \[ \frac{1}{3} x = 0 \] \( x = 0 \). Thus, the zero of \( f \) is \( x = 0 \).

Key Concepts

Evaluating Functions

When dealing with functions, it's crucial to know how to evaluate them for different values of \( x \). This simply means substituting a given value of \( x \) into the function and calculating the result. Let's take the function \( f(x) = \frac{1}{3}x \). To evaluate \( f(-2) \), substitute \( -2 \) into the equation:

• Replace \( x \) with \( -2 \): \( f(-2) = \frac{1}{3}(-2) \).
• Solve the expression to find that \( f(-2) = -\frac{2}{3} \).

For \( f(4) \), follow a similar process:

• Substitute \( 4 \) into the function: \( f(4) = \frac{1}{3}(4) \).
• This results in \( f(4) = \frac{4}{3} \).

Evaluating functions helps you understand how the output value \( f(x) \) changes with different inputs.

Graphing Functions

Graphing a function transforms an algebraic equation into a visual representation on a coordinate plane. Our function \( f(x) = \frac{1}{3}x \) is a linear function, meaning its graph will be a straight line. This is because linear functions always have the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. For our function:

• The slope \( m \) is \( \frac{1}{3} \), indicating that for each unit increase in \( x \), \( f(x) \) increases by \( \frac{1}{3} \).
• The graph passes through the origin (0,0) since the y-intercept \( b \) is 0.

The graph of \( f(x) = \frac{1}{3}x \) will be a straight line through these points. By graphing, you can visually assess how the function behaves across different values of \( x \). Also, this allows you to spot critical points like zeros, intercepts, and slopes.

Zeros of Functions

The zero of a function is the point or points where the function's graph intersects the x-axis. This occurs where \( f(x) = 0 \). For the function \( f(x) = \frac{1}{3}x \), algebraically, we find zeros by setting the equation to zero:

• \( \frac{1}{3}x = 0 \)
• Solving for \( x \) gives \( x = 0 \).

Thus, the zero of the function is \( x = 0 \), meaning the graph crosses the x-axis at the origin. While the algebraic approach precisely identifies zeros, graphing provides a visual confirmation.Knowing the zeros of a function is important because it shows where the output changes sign, transitioning from positive to negative or vice versa. In practical applications, zeros can represent crucial points in various contexts, such as profit break-evens in economics or turning points in physics.