Problem 20 - 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 \(f\) $$f(x)=\frac{1}{2} x-2$$

Step-by-Step Solution

Answer

The zero of the function is 4.

Step 1: Evaluate f(-2)

To find \(f(-2)\), substitute \(-2\) into the function \(f(x) = \frac{1}{2}x - 2\). Calculate as follows: \[ f(-2) = \frac{1}{2}(-2) - 2 = -1 - 2 = -3 \] Hence, \(f(-2) = -3\).

Step 2: Evaluate f(4)

To find \(f(4)\), substitute \(4\) into the function \(f(x) = \frac{1}{2}x - 2\). Calculate as follows: \[ f(4) = \frac{1}{2}(4) - 2 = 2 - 2 = 0 \] Hence, \(f(4) = 0\).

Step 3: Graph the function

The function \(f(x) = \frac{1}{2}x - 2\) is a linear function with a slope of \(\frac{1}{2}\) and a y-intercept of \(-2\). Begin plotting the graph at the y-intercept \((0, -2)\). Then, use the slope to find another point: from \((0, -2)\), move up 1 unit and right 2 units to reach the point \((2, -1)\). Draw a line through these points.

Step 4: Graph zero of the function

The zero of the function is where the graph intersects the x-axis, i.e., where the output \(f(x) = 0\). In this exercise, \(f(4) = 0\), so the graph will cross the x-axis at \(x = 4\).

Step 5: Calculate the zero algebraically

To find the zero of \(f\) algebraically, set the function equal to zero and solve for \(x\): \[ 0 = \frac{1}{2}x - 2 \] Add 2 to both sides: \[ 2 = \frac{1}{2}x \] Multiply both sides by 2 to solve for \(x\): \[ x = 4 \] Thus, the zero of \(f\) is \(x = 4\).

Key Concepts

Function Evaluation

When we talk about evaluating a function, we are discussing substituting a given value for the variable in the function and performing the arithmetic to find the function's output. To evaluate a function like the one given, \(f(x) = \frac{1}{2}x - 2\), you simply substitute the given values into the function. For example:

• To find \(f(-2)\), replace \(x\) with \(-2\). This results in \(f(-2) = \frac{1}{2}(-2) - 2 = -3\).


• Similarly, to evaluate \(f(4)\), substitute \(4\) for \(x\), giving you \(f(4) = 0\).

Function evaluation helps us determine specific points on the graph of the function, which can be crucial in understanding the behavior of the function.

Graphing Linear Functions

Graphing a linear function involves plotting points and drawing a straight line through them. For the function \(f(x) = \frac{1}{2}x - 2\), you start by identifying two key components:

• The slope, which is \(\frac{1}{2}\).


• The y-intercept, which is -2 (the point where the line crosses the y-axis).

Start plotting by placing a point at the y-intercept (0, -2). From here, use the slope to determine another point. The slope tells us to go up 1 unit and 2 units to the right from the y-intercept, which lands us at (2, -1).Connecting these points with a straight line gives the complete graph of the function. Graphing helps visualize the solution and provides insights into the function's behavior and characteristics, such as identifying intercepts and trends.

Finding Zeros of Functions

Finding the zeros of a function involves determining the input value that makes the output of the function zero. This is important because the zero represents the point where the graph of the function crosses the x-axis.For the function \(f(x) = \frac{1}{2}x - 2\), finding the zero means solving for \(x\) when \(f(x) = 0\):\[0 = \frac{1}{2}x - 2\]Add 2 to both sides to isolate the term with \(x\):\[2 = \frac{1}{2}x\]Then multiply by 2:\[x = 4\]Thus, the function's zero is \(x = 4\), meaning the graph crosses the x-axis at this point.Understanding how to find zeros is crucial for understanding the roots or solutions of equations, thereby solving real-world problems involving linear functions.