Problem 93 - Applied Calculus

Question

If a linear function is such that \(f(4)=7\) and \(f(6)=11,\) then \(f(5)=?\) [Hint: No work necessary.]

Step-by-Step Solution

Answer

The value of \(f(5)\) is 9.

Step 1: Identify the Linear Function Form

A linear function can be written as \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. Our task is to find the value of this function at \(x = 5\).

Step 2: Use Given Points to Calculate Slope

We are given two points on the line: \((4, 7)\) and \((6, 11)\). The slope \(m\) is calculated as \(m = \frac{11 - 7}{6 - 4} = \frac{4}{2} = 2\).

Step 3: Write the Function Using the Slope and a Point

Using the slope \(m = 2\) and one of the points, such as \(f(4) = 7\), we can find \(b\) by substituting into the function: \(7 = 2 \times 4 + b\). Solving for \(b\), we get \(b = -1\). The equation of the function is \(f(x) = 2x - 1\).

Step 4: Calculate \(f(5)\)

Substitute \(x = 5\) into the function \(f(x) = 2x - 1\): \(f(5) = 2 \times 5 - 1 = 10 - 1 = 9\).

Key Concepts

Slope Calculation

Understanding how to calculate the slope of a linear function is a foundational mathematical skill. The slope, denoted as \( m \), measures the rate of change of one variable relative to another. It tells us how steep or flat a line is. For two points \((x_1, y_1)\) and \((x_2, y_2)\) on a line, the slope is calculated as:

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

In our problem, we are given two points: \((4, 7)\) and \((6, 11)\). Using the formula, the slope \( m \) is:

• \( m = \frac{11 - 7}{6 - 4} = \frac{4}{2} = 2 \)

The result, \( m = 2 \), means that for every unit increase in \( x \), \( y \) increases by 2. This tells us how rapidly \( y \) changes as \( x \) changes on this line.

Point-Slope Form

The point-slope form is a useful way to write the equation of a line when you know one point on the line and the slope. The formula is written as:

• \( y - y_1 = m(x - x_1) \)

Where \( (x_1, y_1) \) is a given point on the line, and \( m \) is the slope. In our exercise, we followed a similar principle by determining the slope \( m = 2 \), and using a known point \((4, 7)\) to find the equation of the line.By substituting into the point-slope form we used:

• \( y - 7 = 2(x - 4) \)

Simplifying it to the slope-intercept form (which is more common):

• \( y = 2x - 1 \)

This transformation from point-slope to slope-intercept form allows for easier calculation of any other \( y \) values by simply substituting \( x \).

Y-Intercept

The y-intercept is where the graph of a linear equation crosses the y-axis. It's a crucial part of the linear equation in slope-intercept form: \( y = mx + b \). The \( b \) in this formula represents the y-intercept.To find the y-intercept within our example, we had already calculated the slope \( m = 2 \), and used the point \((4, 7)\) for our calculations:

• Using the formula: \( 7 = 2 \times 4 + b \)
• We solve for \( b \): \( 7 = 8 + b \)
• Thus, \( b = 7 - 8 = -1 \)

The y-intercept \( b = -1 \) indicates that the line crosses the y-axis at \( (0, -1) \). This is a vital point for sketching the graph of the equation and analyzing the function.