Problem 14 - 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 \(f\) $$f(x)=-3 x$$

Step-by-Step Solution

Answer

(a) \( f(-2) = 6 \), \( f(4) = -12 \). (b) The graph intersects the x-axis at the origin, showing the zero is 0. (c) The zero of \( f \) is 0.

Step 1: Identify the Function

We are given the linear function \( f(x) = -3x \). A linear function is expressed in the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. In this case, \( m = -3 \) and \( b = 0 \).

Step 2: Evaluate f(-2)

Substitute \( x = -2 \) into the function. Calculate \( f(-2) = -3(-2) \). Since \( -3 \times -2 = 6 \), we find \( f(-2) = 6 \).

Step 3: Evaluate f(4)

Substitute \( x = 4 \) into the function. Calculate \( f(4) = -3(4) \). Since \( -3 \times 4 = -12 \), we find \( f(4) = -12 \).

Step 4: Graph the Function

To graph \( f(x) = -3x \), we plot the points \((-2, 6)\) and \((4, -12)\) and draw the line through them. This line will slope downwards, moving from the top left towards the bottom right, as the slope \( m = -3 \) is negative and the y-intercept \( b = 0 \) is at the origin.

Step 5: Determine the Zero from the Graph

The zero of the function is where the graph intersects the x-axis. Since the y-intercept is at the origin \((0, 0)\), this point is also the zero of the function.

Step 6: Find the Zero Algebraically

To find the zero algebraically, set \( f(x) = 0 \) and solve for \( x \). This gives \(-3x = 0\). Solving for \( x \), we divide both sides by \(-3\), obtaining \( x = 0 \). The zero of the function is \( x = 0 \).

Key Concepts

Graphing Linear Equations

Graphing linear equations helps us understand the behavior of linear functions visually. In our example, the function given is \( f(x) = -3x \). When graphing a linear equation, it's essential to plot at least two points to determine the line. For \( f(x) \), consider the points \((-2, 6)\) and \((4, -12)\). These points can be plotted on the graph, and the resulting line will pass through them.
This line will slope downwards from the top left to the bottom right, demonstrating a negative slope. Since the slope is \( -3 \), for every one unit increase in \( x \), \( f(x) \) will decrease by three units, giving us a visual representation of the function's gradient.

Slope-Intercept Form

The slope-intercept form of a linear equation is a straightforward way to express a linear function. It is given by \( f(x) = mx + b \), where \( m \) represents the slope and \( b \) is the y-intercept.
For the function \( f(x) = -3x \), the slope \( m \) is \(-3\), indicating the line goes down as you move to the right. The y-intercept \( b \) is 0 in this case, so the line crosses the y-axis at the origin. Understanding this form makes it much easier to visualize and graph linear functions without needing to calculate several points.

Finding Zeros of Functions

Finding zeros of a function means identifying points where the output is zero. In graphical terms, this is where the function crosses the x-axis. For the function \( f(x) = -3x \), the graph intersects the x-axis at the origin, \( (0, 0) \).
To find the zero algebraically, you set \( f(x) \) equal to zero and solve for \( x \). This gives the equation \( -3x = 0 \). Dividing both sides by \(-3\) results in \( x = 0 \). Thus, the zero of the function is at \( x = 0 \). Zeros are crucial as they reveal the roots of the equation.

Evaluating Functions

Evaluating a function involves substituting a specific value for \( x \) into the function and calculating the result. For instance, to evaluate \( f(x) = -3x \) at \( x = -2 \), substitute \(-2\) into the equation resulting in \( f(-2) = -3(-2) = 6 \).
This means when \( x \) is \(-2\), the function outputs 6. Similarly, for \( x = 4 \), we compute \( f(4) = -3 \times 4 = -12 \). Evaluating functions helps to determine specific points that lie on the graph, which in turn aids in graphing more accurately.