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

Step-by-Step Solution

Answer

(a) \(f(-2) = 8\), \(f(4) = -10\); (b) Graph intersects x-axis at \((\frac{2}{3}, 0)\); (c) Zero is \(x = \frac{2}{3}\).

Step 1: Evaluate f(-2)

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

Step 2: Evaluate f(4)

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

Step 3: Graph f(x)

The graph of the function \(f(x) = -3x + 2\) is a straight line. Start by identifying two points from the calculated values: \((-2, 8)\) and \((4, -10)\). Plot these points on the coordinate plane. Draw a line through the points, extending in both directions. This line is the graph of \(f\).

Step 4: Use the Graph to Find the Zero

The zero of a function \(f\) is the point where the graph intersects the x-axis. This value is the x-intercept of the line. On the graph, find where the line crosses the x-axis, which is where \(f(x) = 0\).

Step 5: Calculate the Zero Algebraically

Set \(f(x) = 0\) to find the zero of the function. Substitute into the equation:\[-3x + 2 = 0.\]Solve for \(x\):\[-3x = -2\]\[x = \frac{2}{3}.\]Therefore, the zero of \(f\) is \(x = \frac{2}{3}\).

Key Concepts

Function Evaluation

Evaluating a function means finding the value of the function for a specific value of the independent variable, typically represented by \( x \). For linear functions like \( f(x) = -3x + 2 \), this process involves substitution and simplification.

• To evaluate \( f(-2) \), substitute \( x = -2 \) into the function: \( f(-2) = -3(-2) + 2 \).
• Calculate: \( -3(-2) = 6 \). Add 2 to get \( f(-2) = 8 \).

Similarly, to evaluate \( f(4) \):

• Substitute \( x = 4 \) in the function, yielding \( f(4) = -3(4) + 2 \).
• Perform the arithmetic: \( -3 \times 4 = -12 \) plus 2 gives \( f(4) = -10 \).

Function evaluation is useful to determine function output for specific inputs. It is crucial for verifying predictions based on function behavior.

Graphing Linear Equations

Graphing linear equations like \( f(x) = -3x + 2 \) involves plotting points and drawing a straight line.First, use evaluated points such as \((-2, 8)\) and \((4, -10)\) from function evaluations.

• Locate \((-2, 8)\) on the coordinate plane: \(-2\) on the x-axis and 8 on the y-axis.
• Locate \((4, -10)\) similarly: 4 on the x-axis and -10 on the y-axis.

Once both points are plotted:

• Draw a straight line through them. This line extends infinitely in both directions.
• The slope is \(-3\), representing the rate of change. For every unit increase in \( x \), \( y \) decreases by 3.

Graphing helps visualize solutions of the equations and relationships among variables. It offers a geometric interpretation of linear equations.

Finding Zeros of Functions

Finding the zero of a function means determining where the function equals zero. For a linear function \( f(x) = -3x + 2 \), the zero corresponds to the x-coordinate of the x-intercept of the graph.To find the zero:

• Set \( f(x) = 0 \): \(-3x + 2 = 0\).
• Solve for \( x \) by isolating it: \(-3x = -2\).
• Divide by \(-3\): \( x = \frac{2}{3} \). This is the zero of the function.

Graphically, the zero is where the line crosses the x-axis. This point, \( x = \frac{2}{3} \), represents where the output of the function changes from positive to negative or vice versa. Knowing the zeros is essential for understanding function behavior and for solving equations in many applications.