Problem 8 - College Algebra with Modeling and Visualization

Question

Exercises \(7-10:\) Find the formula for a linear function \(f\) that models the data in the table exactly. $$ \begin{array}{rlrl} x & -6 & 0 & 3 \\ f(x) & -5 & -1 & 1 \end{array} $$

Step-by-Step Solution

Answer

The linear function is \(f(x) = \frac{2}{3}x - 1\).

Step 1: Identify Two Points

From the table, choose two points to determine the linear function. Let's use the points \((-6, -5)\) and \((0, -1)\).

Step 2: Calculate the Slope

To find the slope \(m\) of the linear function, use the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substitute the points \((x_1, y_1) = (-6, -5)\) and \((x_2, y_2) = (0, -1)\), resulting in: \(m = \frac{-1 - (-5)}{0 - (-6)} = \frac{4}{6} = \frac{2}{3}\).

Step 3: Use Point-Slope Form

With the slope found, use the point-slope form of a linear equation to find the function equation: \(y - y_1 = m(x - x_1)\). Substitute \(m = \frac{2}{3}\) and \((x_1, y_1) = (0, -1)\): \(y + 1 = \frac{2}{3}(x - 0)\).

Step 4: Simplify to Slope-Intercept Form

Simplify the equation from Step 3 to the slope-intercept form \(y = mx + b\). Start with: \(y + 1 = \frac{2}{3}x\). Subtract 1 from both sides to get \(y = \frac{2}{3}x - 1\).

Step 5: Verify with a Third Point

Verify the linear function \(f(x) = \frac{2}{3}x - 1\) using the third point \((3, 1)\). Substitute \(x = 3\) into the equation: \(f(3) = \frac{2}{3}(3) - 1 = 2 - 1 = 1\). The result matches the provided value, confirming the function is correct.

Key Concepts

Slope Calculation

One of the fundamental steps in working with linear functions is calculating the slope. The slope, often represented by "m", indicates how steep the line is. It tells us how much the value of y changes for a unit change in x. To calculate the slope, we use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula takes two points from the line,

• Point 1: \((x_1, y_1)\)
• Point 2: \((x_2, y_2)\)

By subtracting the y-values and dividing by the differences between the x-values, you get the slope. In our example, using the points \((-6, -5)\) and \((0, -1)\), the calculation becomes:\[ m = \frac{-1 - (-5)}{0 - (-6)} = \frac{4}{6} = \frac{2}{3} \]This shows that for every increase of 3 units in x, y increases by 2 units. This positive slope indicates an upward trend.

Point-Slope Form

The point-slope form is a useful way to write the equation of a line when you know one point and the slope of the line. It is particularly handy for building the equation step by step. The point-slope form is given by the equation: \[ y - y_1 = m(x - x_1) \]Where:

• \( m \) is the slope we calculated previously.
• \((x_1, y_1)\) are the coordinates of the known point.

Using our known point \((0, -1)\) and the slope \(\frac{2}{3}\), the equation becomes:\[ y + 1 = \frac{2}{3}(x - 0) \]This form allows easy transition to other forms and can help verify if additional points lie on the line by substituting their coordinates.

Slope-Intercept Form

The slope-intercept form is one of the most straightforward ways to express linear functions and is widely used due to its simplicity. The equation follows the format:\[ y = mx + b \]Here,

• \( m \) represents the slope.
• \( b \) stands for the y-intercept, the point where the line crosses the y-axis.

To convert our point-slope equation into this form, simplify it:From \( y + 1 = \frac{2}{3}x \), subtract 1 from both sides:\[ y = \frac{2}{3}x - 1 \]This final equation is easy to interpret. You can immediately see that the line has a slope of \(\frac{2}{3}\) and crosses the y-axis at \(-1\). It provides a quick way to sketch the graph or understand the relationship between the variables.