Problem 14 - A Graphical Approach to College Algebra

Question

bWork 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 \(f\) $$f(x)=-3 x$$

Step-by-Step Solution

Answer

(a) \(f(-2) = 6\), \(f(4) = -12\); (b) Graph is a line crossing origin; (c) Zero is \(x = 0\).

Step 1: Identify the Function

The given linear function is \(f(x) = -3x\). Our task involves evaluating the function at specific points, graphing it, and finding its zero.

Step 2: Evaluate f(-2)

Substitute \(x = -2\) into the function: \(f(-2) = -3(-2) = 6\).

Step 3: Evaluate f(4)

Substitute \(x = 4\) into the function: \(f(4) = -3(4) = -12\).

Step 4: Graph the Function

Plot the function \(f(x) = -3x\) on the coordinate plane. Since it's a linear function, it will be a straight line passing through the origin (0,0) with a slope of -3.

Step 5: Relate Graph to Zero of the Function

The graph of \(f(x) = -3x\) intersects the x-axis at the zero of the function. This intersection point is where the value of \(f(x)\) equals zero.

Step 6: Find the Zero of f

To find the zero of \(f(x) = -3x\), set the function equal to zero: \(-3x = 0\). Solving for \(x\) gives \(x = 0\).

Key Concepts

Evaluate Function

Evaluating a function essentially means determining the output or value of the function for specific input values. For the linear function given in the exercise, which is \(f(x) = -3x\), we need to substitute the given values into the equation to find the corresponding outputs.

Here's how we do it:

• First, we evaluate \(f(-2)\). To achieve this, substitute \(x = -2\) into the function to get \(f(-2) = -3(-2)\). By simplifying this expression, you get \(f(-2) = 6\).
• Next, evaluate \(f(4)\) by substituting \(x = 4\) into the function: \(f(4) = -3(4)\). Simplifying gives \(f(4) = -12\).

By performing these substitutions, you can efficiently find the value of the function at any specific point. This process is essential in analyzing how a linear function behaves across its domain.

Graph Function

Graphing a function is a visual way to represent all possible inputs and their corresponding outputs. For a linear function like \(f(x) = -3x\), the graph will be a straight line. This straight line represents all the solutions to the linear equation.

To graph \(f(x) = -3x\):

• Identify the y-intercept. In this case, the y-intercept is at the origin, (0,0), because the function does not have a constant term.
• The slope of the line is -3. This means that for every 1 unit you move horizontally to the right, you move 3 units down since the slope is negative.
• By plotting the points, beginning with (0,0) and using the slope, you'll draw a line through these points.

Graphing not only helps us visualize the relationship between x and y but also assists in determining other characteristics of the function, such as its zero.

Zero of a Function

The zero of a function is the value of \(x\) that makes the function value \(f(x)\) equal zero. It's the point where the graph of the function intersects the x-axis. For linear functions like \(f(x) = -3x\), finding the zero is straightforward.

Here's how you determine it:

• To find the zero, set the function equal to zero: \(-3x = 0\).
• Solve the equation for \(x\). In this case, dividing both sides by -3, you get \(x = 0\).
• The zero of the function is thus \(x = 0\), which corresponds to the x-coordinate of the place where the graph crosses the x-axis.

Knowing the zero of a function is crucial because it provides insight into the root or solution of the function equation and it aids in understanding the behavior of the graph in various mathematical and real-life contexts.