Problem 14 - 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. \(f(3) = 9\), \(f(-1) = -11\)

Step-by-Step Solution

Answer

The linear function that satisfies the given conditions is \(f(x) = 5x - 6\) and the graph is a straight line with a slope of 5 and y-intercept of -6.

Step 1: Calculate the slope(m)

The slope can be calculated by using the formula \((y_2 - y_1) / (x_2 - x_1)\). Here, \(x_1 = 3\), \(y_1 = 9\), \(x_2 = -1\) and \(y_2 = -11\). By substituting these values in the formula, we get \(m = (-11 - 9) / (-1 - 3) = 20 / 4 = 5\). Hence, the slope is 5.

Step 2: Find the y-intercept(c)

The y-intercept can be calculated using the formula \(y = mx + c\). Rearranging for c, we get \(c = y - mx\). Substituting one of the points, let's say (3, 9), and the slope into the formula, we obtain \(c = 9 - 5 * 3 = 9 - 15 = -6\). So, the y-intercept is -6.

Step 3: Write the linear function

Now that we have the slope (5) and the y-intercept (-6), we can write the linear function as \(f(x) = 5x - 6\). This function satisfies the given conditions \(f(3) = 9\) and \(f(-1) = -11\).

Step 4: Sketch the function

Start with the y-intercept, -6, as the point where the line crosses the y-axis. The slope of 5 means that for every 1 unit moved to the right, the line moves up by 5 units. Plotting this on a grid will give the graph of the function \(f(x) = 5x - 6\).

Key Concepts

Slope Calculation

The concept of slope is essential in understanding linear functions. The slope tells us how steep a line is, and it is calculated as the ratio of vertical change to horizontal change between two points. To find this, we use the equation:

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

Using the given points

• \((x_1, y_1) = (3, 9)\)
• \((x_2, y_2) = (-1, -11)\)

we substitute them into the formula. Therefore:
\[ m = \frac{-11 - 9}{-1 - 3} = \frac{20}{4} = 5 \]
The slope \(m\) here is 5, showing a fairly steep upward slant from left to right. This means for every unit you move to the right along the x-axis, you move up 5 units along the y-axis.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. It's crucial because it gives a starting point for graphing the function. To find it, use the slope-intercept form of a linear equation:

• \( y = mx + c \)

where \(c\) represents the y-intercept. Rearrange this formula to solve for \(c\), leading to:
\[ c = y - mx \]Using one of the known points, say

• \((3,9)\)

with a slope \(m = 5\), calculate \(c\) as follows:
\[ c = 9 - 5 \times 3 = 9 - 15 = -6 \]
Thus, the y-intercept is \(-6\), which means the line crosses the y-axis at the point (0,-6). This is a key point to start sketching the graph.

Graph Sketching

Sketching the graph of a linear function involves plotting points based on the slope and y-intercept. For the function \(f(x) = 5x - 6\), start at the y-intercept:

• Point (0,-6)

This is where the line meets the y-axis.
Next, use the slope to determine direction and steepness. A slope of 5 means that for every 1 unit you move to the right, the line rises by 5 units. This gives you the next point

• (1, -1) when moved from (0, -6)

Draw the line through these points extending on the grid.
This visualization helps understand how the function behaves and allows checking if other function values are correctly placed on the line. For example, our calculated line aligns with the points (3,9) and (-1,-11), confirming correctness.