Problem 17 - 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 \(f\) $$f(x)=\frac{2-x}{4}$$

Step-by-Step Solution

Answer

(a) f(-2) = 1, f(4) = -1/2. (b) The graph crosses the x-axis at x = 2. (c) The zero is x = 2.

Step 1: Substitute and Evaluate f(-2)

Substitute \(-2\) for \(x\) in the function: \[f(-2) = \frac{2 - (-2)}{4} = \frac{2 + 2}{4} = \frac{4}{4} = 1\]

Step 2: Substitute and Evaluate f(4)

Substitute \(4\) for \(x\) in the function: \[f(4) = \frac{2 - 4}{4} = \frac{-2}{4} = -\frac{1}{2}\]

Step 3: Sketch the Graph of f

Plot the points \((-2, 1)\) and \((4, -\frac{1}{2})\) on the graph. Connect them with a line to represent the graph of the function \(f(x) = \frac{2-x}{4}\). This is a linear function and will appear as a straight line.

Step 4: Determine the Zero Using the Graph

The zero of the function \(f\) occurs where the function crosses the x-axis. Observe the graph and find the x-value where \(f(x) = 0\).

Step 5: Algebraically Calculate the Zero

Set \(f(x)\) equal to 0 and solve for \(x\): \[0 = \frac{2-x}{4} \]Multiply both sides by 4 to clear the fraction: \[0 = 2 - x \]Solving for \(x\), we get:\[x = 2\]

Step 6: Verify the Zero

Substitute \(2\) back into the original function to verify:\[ f(2) = \frac{2-2}{4} = \frac{0}{4} = 0\]Thus, x = 2 is indeed the zero of the function.

Key Concepts

Evaluating Functions

Understanding how to evaluate functions is crucial in mathematics. A function can be thought of as a special machine that gives you output after processing an input. For a given function like \(f(x) = \frac{2-x}{4}\), evaluating it involves substituting a specific value for \(x\). This helps us determine what the function spits out based on the input value.

• Step-by-step Evaluation: To find \(f(-2)\), we substitute \(-2\) into the function, processing it as follows: \(f(-2) = \frac{2 - (-2)}{4} = \frac{4}{4} = 1\). Similarly, for \(f(4)\), substitute 4: \(f(4) = \frac{2 - 4}{4} = -\frac{1}{2}\).


• Why Evaluate? Evaluating helps to understand the behavior of the function at specific points, which is vital when analyzing or using the function for other applications.

Graphing Functions

Graphing a linear function like \(f(x) = \frac{2-x}{4}\) involves plotting points on a coordinate plane and connecting these points to form a straight line.

• Plotting Points: Start with evaluating the function at notable points, like those found earlier:
  • \((-2, 1)\)
  • \((4, -\frac{1}{2})\)
  These are coordinates that help establish the placement of the line on the graph.


• Drawing the Line: Connect the plotted points with a straight line. This line visually represents the function and provides an immediate visual insight into its behavior across different sections of the x-axis.

Graphing functions gives a clear visual of where the function crosses the axes, reflects symmetry, and shows trends like increasing or decreasing behavior.

Finding Zeros of a Function

The zero of a function is the \(x\)-value where the function's graph crosses the x-axis. For these points, the output \(f(x)\) is zero.

• Using the Graph: Look for where the line you've plotted crosses the x-axis. This point is the zero of the function. In our earlier exercise, the graph helps us identify this point quickly.


• Importance of Zeros: Finding zeros is critically important because they often represent solutions to equations modeled by functions, provide points of interest which can reveal significant insights into the function's behavior, and are useful in a variety of applications ranging from root-finding algorithms to complex systems modeling.

Algebraic Solutions

Solving a function algebraically involves finding an unknown value that satisfies the function equation, typically done without graphing.

• Setting the Function to Zero: To find the zero algebraically, set \(f(x) = 0\) and solve for \(x\). For \(f(x) = \frac{2-x}{4}\), the equation becomes: \[0 = \frac{2-x}{4}\]. Multiply by 4 to eliminate the denominator: \[0 = 2 - x\].
• Solve the Equation: Solve for \(x\) by isolating it: \(x = 2\). This solution proves \(x = 2\) is where the function crosses the x-axis.

This approach confirms the graphical intersection point algebraically, strengthening our overall understanding and verifying the graph's visual evidence with direct calculation.