Problem 12 - 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 \(f\) $$f(x)=\frac{1}{4} x+\frac{1}{2}$$

Step-by-Step Solution

Answer

(a) \(f(-2) = 0, f(4) = \frac{3}{2}\). (b) Use the line to find the x-intercept (zero). (c) Zero is \(x = -2\).

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

First, we need to substitute the given values into the function to find the desired outputs. To evaluate \(f(-2)\), substitute \(-2\) into the equation as follows:\[f(-2) = \frac{1}{4} (-2) + \frac{1}{2} = -\frac{1}{2} + \frac{1}{2} = 0\]Next, evaluate \(f(4)\) by substituting \(4\) into the equation:\[f(4) = \frac{1}{4} (4) + \frac{1}{2} = 1 + \frac{1}{2} = \frac{3}{2}\]Thus, \(f(-2) = 0\) and \(f(4) = \frac{3}{2}\).

Step 2: Graph f(x) = (1/4)x + (1/2)

To graph the function \(f(x)=\frac{1}{4}x+\frac{1}{2}\), we start by plotting the y-intercept, which is \((0, \frac{1}{2})\). Next, use the slope, which is \(\frac{1}{4}\), indicating a rise of 1 unit for every 4 units run to plot another point starting from the y-intercept. For example, the point \((4, \frac{3}{2})\) can be used, as calculated previously.Draw a straight line through these points to represent the function. The zero of the function is where the line crosses the x-axis.

Step 3: Use the Graph to Determine the Zero

The zero of a function corresponds to the x-value where the function crosses the x-axis (i.e., \(f(x) = 0\)). From the graph of \(f(x) = \frac{1}{4}x + \frac{1}{2}\), we can see the function crosses the x-axis at \(x = -2\). This confirms our previous finding that \(f(-2) = 0\).

Step 4: Algebraically Find the Zero

Set the function equal to zero to determine the x-value where the function crosses the x-axis.\[0 = \frac{1}{4}x + \frac{1}{2}\]Subtract \(\frac{1}{2}\) from both sides:\[-\frac{1}{2} = \frac{1}{4}x\]Multiply both sides by \(4\) to solve for \(x\):\[x = -2\]So, the zero of the function is indeed \(x = -2\).

Key Concepts

Graphing Linear Equations

Graphing linear equations is an essential skill in understanding linear functions. A linear equation can be represented graphically as a straight line. The first step is to identify key components of the equation, mainly the slope and the y-intercept. For a linear function like \(f(x) = \frac{1}{4}x + \frac{1}{2}\), it's in the slope-intercept form, which we'll discuss in detail later on.

When graphing, start by plotting the y-intercept. For our function, the y-intercept is \((0, \frac{1}{2})\). This is the point where the line crosses the y-axis.

• Plot the y-intercept as your first point on the graph.
• Use the slope to plot additional points.
• Slope of \(\frac{1}{4}\) means rise of 1 unit for every 4 units of horizontal run.

For example, from the y-intercept, move up 1 unit and 4 units to the right to plot another point, resulting in \((4, \frac{3}{2})\). Draw a line through these points to extend the line across the grid. The beauty of linear functions is their consistency; once you have two points, the entire line is determined.

Finding Function Zeros

Finding the zero of a function is like finding a treasure map's X mark. It's the point where the function crosses the x-axis, representing the solution where the function equals zero. For our function \(f(x) = \frac{1}{4}x + \frac{1}{2}\), the zero is the x-value where \(f(x) = 0\).

To find the zero graphically, look at where the line intersects the x-axis on the graph. In this case, it crosses at \(x = -2\). You can verify this zero algebraically by setting the function equal to zero and solving for \(x\).

• Set \(f(x) = 0\): \(0 = \frac{1}{4}x + \frac{1}{2}\).
• Subtract \(\frac{1}{2}\) from both sides: \(-\frac{1}{2} = \frac{1}{4}x\).
• Multiply both sides by 4: \(x = -2\).

Therefore, \(x = -2\) is the zero of the function, and both graphing and algebraic methods confirm this solution.

Slope-Intercept Form

The slope-intercept form of a linear equation is an accessible format for analyzing and graphing linear functions. It is written as \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept. This form provides key information at first glance:

• The slope \(m\) determines the steepness and direction of the line. A positive slope means the line rises from left to right. A negative slope means it falls.
• The y-intercept \(b\) indicates where the line crosses the y-axis, which is straightforward to plot.

For our function, \(f(x) = \frac{1}{4}x + \frac{1}{2}\), \(m = \frac{1}{4}\) and \(b = \frac{1}{2}\). This tells us the line rises slowly (since \(\frac{1}{4}\) is a gentle slope) and starts at \((0, \frac{1}{2})\) on the y-axis. Understanding the slope-intercept form simplifies the process of plotting the graph and analyzing the function, making it an invaluable tool for any math student dealing with linear functions.