Problem 9 - 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 \(f\) $$f(x)=x+2$$

Step-by-Step Solution

Answer

(a) \(f(-2) = 0\), \(f(4) = 6\); (b) Zero occurs at \(x = -2\); graph determines zero when intersecting x-axis; (c) zero is \(x = -2\).

Step 1: Evaluate f(-2)

To evaluate \(f(-2)\), substitute \(x = -2\) into the function \(f(x) = x + 2\). This gives us \(f(-2) = -2 + 2 = 0\).

Step 2: Evaluate f(4)

To evaluate \(f(4)\), substitute \(x = 4\) into the function \(f(x) = x + 2\). This gives us \(f(4) = 4 + 2 = 6\).

Step 3: Graph the Function

To graph the function \(f(x) = x + 2\), plot the points \((-2, 0)\) and \((4, 6)\) which were calculated in the previous steps. These points can be connected with a straight line.The graph of \(f(x)\) is a straight line with a slope of 1 and a y-intercept at \( (0, 2) \).

Step 4: Determine Zero of f from the Graph

The zero of \(f\) is the x-coordinate where the graph intersects the x-axis. Since we found \(f(-2) = 0\) earlier, the graph intersects the x-axis at \(x = -2\).

Step 5: Calculate the Zero Analytically

To find the zero analytically, set \(f(x) = 0\) and solve for \(x\). \[0 = x + 2\]\[x = -2\]Thus, the zero of \(f\) is \(x = -2\).

Key Concepts

Evaluating Functions

Evaluating a function is all about finding the output, given an input. Essentially, you are substituting a specific value into the function and performing the arithmetic operation. For example, if you have a function like \( f(x) = x + 2 \), to evaluate \( f(-2) \), you substitute \(-2\) for \(x\) in the function:

• Substitute: \( f(-2) = -2 + 2 \)
• Compute: \( f(-2) = 0 \)

This tells us that when the input is \(-2\), the output is 0.
Similarly, to evaluate \( f(4) \):

• Substitute: \( f(4) = 4 + 2 \)
• Compute: \( f(4) = 6 \)

Here, an input of 4 results in an output of 6. Evaluating functions like this is a straightforward method of understanding how changes in \(x\) affect the function's value.

Graphing Linear Equations

Graphing linear equations helps visually represent the relationship between variables in a function.
For the function \( f(x) = x + 2 \), graphing involves plotting points and drawing a straight line through them. Let's break it down:

• Start with the equation: \( y = x + 2 \)
• Plot some points, say \((-2, 0)\) and \((4, 6)\), which we found by substituting \(x\) in the evaluating step.
• Draw a line through the points: This line is y-increasing by 1 for every increase of 1 in \(x\) due to the slope \( m = 1 \).

The graph crosses the y-axis at \((0, 2)\), which is the y-intercept.
The straight line serves as a visual tool for understanding linear relationships. Once you draw the graph, you can use it to estimate function values and explore further into properties like slope and intercepts.

Finding Zeros of Functions

Zeros of a function occur when the output value is zero. Essentially, it is the value of \(x\) that makes \(f(x) = 0\). You can find this value through different methods:

• Graphically: If you have the function \( f(x) = x + 2 \) plotted as a line, the zero is where this line crosses the x-axis. In this case, the graph intersects at \(x = -2\).
• Analytically: Solve the equation \(x + 2 = 0\). Simply subtract 2 from both sides to find \(x = -2\).

Finding the zero is helpful because it tells you the point where the function has no effective change; in other words, where the output is zero.
Understanding the zero of a function aids in solving real-world problems, especially those involving equilibrium points or break-even analysis.