Problem 16 - 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 \(f\) $$f(x)=x+0.5$$

Step-by-Step Solution

Answer

f(-2) = -1.5, f(4) = 4.5, zero of f is -0.5.

Step 1: Evaluating f(-2)

To find \( f(-2) \), substitute \( x = -2 \) into the function \( f(x) = x + 0.5 \). This gives us \( f(-2) = -2 + 0.5 = -1.5 \).

Step 2: Evaluating f(4)

Similarly, substitute \( x = 4 \) into the function \( f(x) = x + 0.5 \). This results in \( f(4) = 4 + 0.5 = 4.5 \).

Step 3: Graphing f(x)

The function \( f(x) = x + 0.5 \) is a linear function, and its graph is a straight line. The y-intercept is \( (0, 0.5) \). Plot two points, such as \( (-2, -1.5) \) from Step 1 and \( (4, 4.5) \) from Step 2, and draw the line through these points.

Step 4: Using the Graph to Find the Zero

The zero of the function is where the graph intersects the x-axis (where \( f(x) = 0 \)). From the graph, identify this point as \( (-0.5, 0) \).

Step 5: Algebraically Finding the Zero

To find the zero algebraically, set \( f(x) = x + 0.5 \) equal to zero: \[ x + 0.5 = 0 \] Solve for \( x \) to find the zero: \[ x = -0.5 \].

Key Concepts

Graphing Linear Equations

Graphing linear equations allows us to visually interpret the relationship between the variables involved. A linear equation represents a straight line, which is why understanding its components is crucial.

• The standard form of a linear equation is typically given as \( y = mx + b \). Here, \( m \) represents the slope, which indicates the steepness of the line. A positive slope means the line rises, whereas a negative slope means it falls. In our function \( f(x) = x + 0.5 \), the slope \( m \) is 1.
• The term \( b \) is the y-intercept, where the line crosses the y-axis. For our function, this is 0.5. So, the line will intersect the y-axis at the point \( (0, 0.5) \).

The straight-line graph can be constructed by plotting points. In this exercise, points \((-2, -1.5)\) and \((4, 4.5)\) derived from evaluating the function help us draw an accurate line.
By understanding the components and plotting systematically, one can easily graph linear functions and use them for analysis.

Finding Zeros

Finding the zeros of a function is about identifying the x-values where the function equals zero. Graphically, this point is where the line crosses the x-axis.

• For the function \( f(x) = x + 0.5 \), finding the zero is crucial as it indicates the x-value where the output \( f(x) \) is 0.
• On a graph, the zero represents the intersection point of the line with the x-axis. For this problem, it's the point \((-0.5, 0)\). This shows that when \( x = -0.5 \), the function becomes zero.

Finding zeros is not just essential for solving mathematical problems but also practical applications like determining break-even points in business.

Evaluating Functions

Evaluating functions means determining the output for specific input values. It's a straightforward process, often involving substituting a number for the variable.

• To evaluate \( f(-2) \) in \( f(x) = x + 0.5 \), replace \( x \) with \(-2\) to get \( f(-2) = -2 + 0.5 = -1.5 \).
• Similarly, for \( f(4) \), substitute \( x = 4 \), resulting in \( f(4) = 4 + 0.5 = 4.5 \).

This aspect helps in determining how a function behaves for given inputs, making it applicable not only in theoretical exercises but in real-world problem-solving as well.