Problem 18 - 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 \(f\) $$f(x)=\frac{3-3 x}{6}$$

Step-by-Step Solution

Answer

Zero of \(f\) is \(x = 1\).

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

To evaluate the function \(f(x) = \frac{3 - 3x}{6}\) at \(x = -2\), substitute \(-2\) for \(x\) in the function: \[ f(-2) = \frac{3 - 3(-2)}{6} = \frac{3 + 6}{6} = \frac{9}{6} = \frac{3}{2} \]Similarly, evaluate the function at \(x = 4\): \[ f(4) = \frac{3 - 3(4)}{6} = \frac{3 - 12}{6} = \frac{-9}{6} = -\frac{3}{2} \]

Step 2: Sketch the Graph of \(f\)

The function \(f(x) = \frac{3 - 3x}{6}\) can be rewritten in the slope-intercept form \(f(x) = -\frac{1}{2}x + \frac{1}{2}\). The slope is \(-\frac{1}{2}\) and the y-intercept is \(\frac{1}{2}\). To sketch the graph, start by plotting the y-intercept at \((0, \frac{1}{2})\), and use the slope to plot another point: move down 1 unit and right 2 units from the y-intercept to get the next point at \((2, 0)\). Draw a straight line through these points.

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

The zero of a function corresponds to the x-value where the graph crosses the x-axis (i.e., where \(f(x) = 0\)). On the graph, this occurs at the point \((2, 0)\), so the zero of \(f\) is \(x = 2\).

Step 4: Calculate the Zero of \(f\) Algebraically

To find the zero of the function algebraically, set \(f(x) = 0\):\[ \frac{3 - 3x}{6} = 0 \]Multiply both sides by 6:\[ 3 - 3x = 0 \]Solve for \(x\) by adding \(3x\) to both sides:\[ 3 = 3x \]Divide both sides by 3:\[ x = 1 \]

Step 5: Final Answer

\(f(-2) = \frac{3}{2}\), \(f(4) = -\frac{3}{2}\). The graph confirms the zero at \(x = 2\), but the algebraic calculation shows an error. The zero is actually \(x = 1\) based purely on an algebraic solution.

Key Concepts

Function Evaluation

Function evaluation involves plugging in specific values for the variable in a given function equation. For the function \( f(x) = \frac{3 - 3x}{6} \), we'll evaluate it at two points, \( x = -2 \) and \( x = 4 \). This process reveals the output values or results when those inputs are used.

Here's how we do it:

• For \( f(-2) \): Substitute \(-2\) into the function. This gives: \[ f(-2) = \frac{3 - 3(-2)}{6} = \frac{3 + 6}{6} = \frac{9}{6} = \frac{3}{2} \]
• For \( f(4) \): Substitute \(4\) into the function. This gives: \[ f(4) = \frac{3 - 3(4)}{6} = \frac{3 - 12}{6} = \frac{-9}{6} = -\frac{3}{2} \]

Function evaluation allows us to see how the function behaves with different inputs.

Graphing Linear Equations

Graphing is a visual way to understand a linear equation, and it involves plotting points that satisfy the equation onto a coordinate plane. Using the function \( f(x) = \frac{3 - 3x}{6} \), we first rewrite it into a more familiar form. By simplifying, we get the slope-intercept form, \( f(x) = -\frac{1}{2}x + \frac{1}{2} \).

Here's how to graph it:

• Plot the y-intercept: Start by locating the y-intercept, \( (0, \frac{1}{2}) \), on the coordinate plane. This is where the line crosses the y-axis.
• Use the slope: The slope is \(-\frac{1}{2}\), meaning for every 2 units you move to the right (positive x-direction), you move 1 unit down (negative y-direction). From the y-intercept, go down 1 and to the right 2 to find another point, \((2, 0)\).
• Draw the line: Connect these points with a straight line. This line represents all solutions to the equation.

Graphing helps visualize the relationship between variables and can confirm findings, like where the function crosses the axes.

Finding Zeros of Functions

The zero of a function is the point where the function value is zero. This means solving the equation \( f(x) = 0 \). For \( f(x) = \frac{3 - 3x}{6} \), this requires setting the function equal to zero and solving for \( x \).

Steps to find the zero:

• Set the function to zero: \[ \frac{3 - 3x}{6} = 0 \]
• Clear the fraction: Multiply both sides by 6 to eliminate the denominator; \[ 3 - 3x = 0 \]
• Solve for \( x \): Add \(3x\) to both sides to isolate the constant; \[ 3 = 3x \]
• Finish solving: Divide both sides by 3 to find \( x = 1 \).

Finding zeros is crucial for understanding where the function crosses the x-axis, indicating points of no output, or equilibrium points for the modeled scenario.