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

Step-by-Step Solution

Answer

(a) \(f(-2) = -\frac{2}{3}\), \(f(4) = \frac{4}{3}\). (b) The zero is where the graph intersects the x-axis. (c) The zero of \(f\) is \(x = 0\).

Step 1: Evaluating f(-2)

To find \(f(-2)\), substitute \(-2\) into the function: \[ f(-2) = \frac{1}{3} \times (-2) = -\frac{2}{3} \]. So, \(f(-2) = -\frac{2}{3}\).

Step 2: Evaluating f(4)

To find \(f(4)\), substitute \(4\) into the function: \[ f(4) = \frac{1}{3} \times 4 = \frac{4}{3} \]. So, \(f(4) = \frac{4}{3}\).

Step 3: Graphing f(x)

The function \(f(x) = \frac{1}{3}x\) is a linear function. To graph, plot the points found in Steps 1 and 2. The y-intercept is at the origin \((0,0)\), as there is no constant term. The slope is \(\frac{1}{3}\), meaning for every 3 units increase in \(x\), \(f(x)\) increases by 1 unit. Draw the line through the points \((-2, -\frac{2}{3})\), \( (0,0) \), and \(( 4, \frac{4}{3})\).

Step 4: Using the Graph to Determine Zero

Since the zero of a function is where it crosses the x-axis, check the graph to see where the line intersects the x-axis. As the graph passes through the point \((0,0)\), the zero of \(f\) is at \(x = 0\).

Step 5: Algebraically Finding the Zero of f(x)

Set the function equal to zero to find where it crosses the x-axis: \[ \frac{1}{3}x = 0 \]. Solving for \(x\), multiply both sides by 3: \(x = 0\). Thus, the zero of \(f\) is indeed \(x = 0\), confirming the graph result.

Key Concepts

Graphing Linear Functions

Graphing a linear function like \( f(x) = \frac{1}{3}x \) involves understanding its structure and using key points to draw it.
A linear function is any function that can be graphed as a straight line. This is because it has the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here, our equation is simplified to \( f(x) = \frac{1}{3}x \), meaning our y-intercept is \( 0 \), and our slope is \( \frac{1}{3} \).

To graph this function:

• Start at the origin (0, 0), because the function simply passes through this point without a constant term to shift it.
• Use the slope \( \frac{1}{3} \). This tells you that from any point on the line, moving 3 units right (increasing \( x \) by 3) results in moving up 1 unit (increasing \( y \) by 1).
• Plot at least two points to establish the line's path - as done in Steps 1 and 2, using the calculations \( (-2, -\frac{2}{3}) \) and \( (4, \frac{4}{3}) \) respectively.
• Connect these points with a straight line extending it across the grid.

By following these steps, you can graph any linear equation, recognizing its direct relationship between \( x \) and \( y \) values.

Finding Zeros of Functions

The zero of a function is crucial; it's where the graph intersects the x-axis.
This point represents where the output \( f(x) \) is zero, indicating that the input \( x \) results in no vertical displacement.

In practical terms, for the function \( f(x) = \frac{1}{3} x \):

• We find the zero by setting the function equal to zero, \( \frac{1}{3}x = 0 \).
• Solve for \( x \): Multiply each side by 3, simplifying to \( x = 0 \).
• This means the function's only zero is at \( x = 0 \).

Visually, on the graph, this zero is evident where the function crosses the x-axis, confirming this analysis. Finding zeros helps you understand where the function changes from positive to negative and vice versa.

Evaluating Functions

Evaluating a function like \( f(x) = \frac{1}{3} x \) involves substituting specific x-values.
This helps in predicting the outputs and understanding the behavior of the function across different inputs.

For our example:

• To find \( f(-2) \), substitute \( -2 \) for \( x \): \[ f(-2) = \frac{1}{3} \times (-2) = -\frac{2}{3} \]. This means if you input \( -2 \), the output is \( -\frac{2}{3} \).
• Similarly, to determine \( f(4) \), replace \( x \) with \( 4 \): \[ f(4) = \frac{1}{3} \times 4 = \frac{4}{3} \]. This indicates that an input of \( 4 \) yields an output of \( \frac{4}{3} \).

This process of substitution and calculation is how functions serve as formulas or rules linking input values to output results.

Slope of a Line

The slope of a line describes its steepness and direction, providing insight into the relationship between the variables involved in a linear function.
For the function \( f(x) = \frac{1}{3} x \), the slope is \( \frac{1}{3} \). Understanding slope involves identifying how changes in \( x \) affect changes in \( y \).

Specifically:

• The slope \( \frac{1}{3} \) indicates that for every three units you move horizontally to the right, the line moves one unit vertically up. This positive slope results in an upward slanting line from left to right.
• The concept of slope is fundamental in determining how quickly \( y \) increases or decreases as \( x \) changes. Here it's a gentle rise, showing a slow increase in value.

The slope helps in predicting future behavior of the linear function and is a critical feature in its graph.