Problem 10 - 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 \(f\) $$f(x)=-3 x+2$$

Step-by-Step Solution

Answer

(a) \(f(-2) = 8\) and \(f(4) = -10\); (b) Graph the function, it crosses the x-axis showing the zero; (c) zero is \(x = \frac{2}{3}\).

Step 1: Evaluate f(-2)

To evaluate \(f(-2)\), substitute \(-2\) into the function \(f(x) = -3x + 2\). \[f(-2) = -3(-2) + 2 = 6 + 2 = 8\] So, \(f(-2) = 8\).

Step 2: Evaluate f(4)

To evaluate \(f(4)\), substitute \(4\) into the function \(f(x) = -3x + 2\). \[f(4) = -3(4) + 2 = -12 + 2 = -10\] So, \(f(4) = -10\).

Step 3: Understand the Graph of f(x)

The linear function \(f(x) = -3x + 2\) can be graphed by plotting two points. We already know two points from steps 1 and 2: \((-2, 8)\) and \((4, -10)\). By connecting these points with a straight line, we create the graph of the function.

Step 4: Use the Graph to Find the Zero of f(x)

The zero of a function \(f(x)\) occurs where the graph crosses the x-axis (where \(f(x) = 0\)). On the graph of \(f(x) = -3x + 2\), find the x-value at this intersection point.

Step 5: Algebraically Find the Zero of f(x)

Set \(f(x) = 0\) and solve for \(x\). \[0 = -3x + 2\] \[3x = 2\] \[x = \frac{2}{3}\]So, the zero of the function is \(x = \frac{2}{3}\).

Key Concepts

Evaluate Function

To evaluate a function means to find the function's output for a particular input value. This involves substituting the input into the function and performing the calculations. For example, given the linear function \( f(x) = -3x +2 \), if you want to evaluate \(f(-2)\), you replace every \(x\) with \(-2\): \[ f(-2) = -3(-2) + 2 \] This simplifies to \(8\), meaning that when \(x = -2\), the output \(f(x)\) is \(8\). Similarly, to evaluate \(f(4)\), substitute \(4\) into the function: \[ f(4) = -3(4) + 2 \] This equals \(-10\). Thus, \(f(4) = -10\), and the output is \(-10\) when \(x = 4\). Evaluating a function is crucial in understanding how changing the input value affects the output. It's a fundamental process in analyzing linear functions.

Graphing Functions

Graphing a function involves plotting its points on a coordinate plane and connecting them to show the relationship between variables. A linear function, like \( f(x) = -3x + 2 \), plots as a straight line. To graph this function, you only need two points. These points can be determined by evaluating the function at different \(x\) values. For instance, from previous evaluations, we have the points \((-2, 8)\) and \((4, -10)\). Plot these on the coordinate plane.

• First point: \((-2, 8)\)
• Second point: \((4, -10)\)

Draw a straight line connecting these points. This line is the visual representation of the function, illustrating the constant rate of change (slope) and intercepts. The slope is \(-3\), indicating the line descends downward from left to right. Graphing helps in visualizing how the function behaves, aiding in tasks like finding zeros or predicting outputs for other inputs.

Zero of the Function

Finding the zero of a function is about identifying where the graph intersects the x-axis. It's the value of \(x\) that makes \(f(x) = 0\). For the linear function \(f(x) = -3x + 2\), you can find this zero both graphically and algebraically.Graphically, look at where the line crosses the x-axis on the graph. This point gives the x-value that corresponds to when \(f(x) = 0\). For the current function, the zero appears where \(x\) is around \(\frac{2}{3}\).Algebraically, find the zero by setting the function equal to zero and solving for \(x\):\[0 = -3x + 2\] \[3x = 2\] \[x = \frac{2}{3}\]So, \(x = \frac{2}{3}\) is the zero of the function. This point is significant as it represents a solution to the equation and reveals the input where the output (\(f(x)\)) is exactly zero. Understanding the zero of a function is essential, particularly in solving equations and interpreting real-world scenarios.