Problem 11 - Precalculus with Limits

Question

In Exercises 11-18, (a) write the linear function \(f\) such that it has the indicated function values and (b) sketch the graph of the function. \(f(1) = 4\), \(f(0) = 6\)

Step-by-Step Solution

Answer

The linear function that satisfies the given conditions is \(f(x) = -2x + 6\). The graph of the function starts at the point (0,6) and has a slope of -2.

Step 1: Finding the Slope

The slope of a line is calculated by the formula \(m = (y_2 - y_1)/(x_2 - x_1)\). Substituting the given points (1, 4) and (0, 6) into the formula gives: \(m = (4 - 6) / (1 - 0) = -2\). Thus, the slope of the function is -2.

Step 2: Determining the y-intercept

The y-intercept of a line is the y-coordinate where the line crosses the y-axis. From the problem, it's known that when \(x = 0\), \(y = 6\), which means the y-intercept is 6.

Step 3: Writing the Function

Using the calculated slope and y-intercept, the linear function can be written as \(f(x) = -2x + 6\)

Step 4: Sketching the Graph

To sketch this function, start by plotting the y-intercept at (0,6). Then, use the slope to determine the next point: from (0,6), move down 2 units and over to the right by 1 which lands at (1, 4). Draw a line through these two points to complete the graph.

Key Concepts

Slope Calculation

Understanding the slope of a line is essential in working with linear functions. The slope (often represented by the letter \(m\)) describes the steepness and direction of a line. For a given linear function, it's calculated using the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

In this exercise, we were given points \((1, 4)\) and \((0, 6)\). By substituting these into the formula, we found the slope:

• \( m = \frac{4 - 6}{1 - 0} = -2 \)

This slope of \(-2\) means the line goes down 2 units for every 1 unit it goes to the right. A negative slope indicates a downward slanting line moving from left to right.
Remember:

• A positive slope means upward as you move right.
• A zero slope is a flat horizontal line.
• An undefined slope is a vertical line.

Y-Intercept

The y-intercept is a crucial part of graphing linear equations. It is the point where the line crosses the y-axis. This means, it's where the x-value is zero. The y-intercept provides a starting spot for graphing a line.
In the problem, when \(x = 0\), \(y = 6\). Therefore, the y-intercept is 6. This tells us that the line will pass through the point \((0, 6)\) on the graph.
This information can be used directly in the equation of the line, giving it a form like:

• \(f(x) = mx + b\)

Where \(b\) is the y-intercept. Thus, our function is \(f(x) = -2x + 6\). Knowing the y-intercept lets you quickly determine the line's position in relation to the y-axis.

Graphing Linear Equations

Graphing linear equations visually represents the relationship between variables. It all starts with understanding the function's form, typically written as \(f(x) = mx + b\).
Once you've calculated the slope and determined the y-intercept, you can easily sketch the graph.To graph the function \(f(x) = -2x + 6\):

• Start at the y-intercept: plot the point \((0, 6)\).
• Use the slope: from \((0, 6)\), move down 2 units and right 1 unit to plot \((1, 4)\).
• Draw a line through these points. The line will extend in both directions.

Graphing linear equations involves these steps:

• Identify the y-intercept on the graph.
• Apply the slope to find another point.
• Connect the points with a straight line.

Visualizing equations helps to grasp the relationship between \(x\) and \(y\) values. This skill is fundamental for understanding real-world linear relationships.