Problem 95 - Brief Applied Calculus

Question

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

Step-by-Step Solution

Answer

To find \(f(5)\) for the linear function, substitute into the equation \(y = 2x - 1\), resulting in \(f(5) = 9\).

Step 1: Understand the Problem

We are given two points on a linear function: \((4, 7)\) and \((6, 11)\). We need to find the value of the function \(f\) at \(x = 5\).

Step 2: Find the Slope of the Line

The slope \(m\) of a line through the points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Using our points \((4, 7)\) and \((6, 11)\), the slope is \(m = \frac{11 - 7}{6 - 4} = \frac{4}{2} = 2\).

Step 3: Write the Equation of the Line

Using the point-slope form of the line equation: \(y - y_1 = m(x - x_1)\), substitute \(m = 2\) and the point \((4, 7)\), we get the equation: \(y - 7 = 2(x - 4)\).

Step 4: Simplify the Line Equation

Simplify the equation \(y - 7 = 2(x - 4)\) to express \(y\) in terms of \(x\). Substituting and rearranging gives: \(y - 7 = 2x - 8\), thus, \(y = 2x - 1\).

Step 5: Calculate the Function Value

Substitute \(x = 5\) into the equation \(y = 2x - 1\) to find \(f(5)\). This gives: \(f(5) = 2(5) - 1 = 10 - 1 = 9\).

Key Concepts

Slope Calculation

The slope of a line is a measure of its steepness and is crucial in understanding linear functions. Think of it as how much the line "rises" vertically for every unit it moves horizontally. The formula to find the slope (\(m\)) between two points (\((x_1, y_1)\)) and (\((x_2, y_2)\)) is:

• \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Here, the change in y (\(y_2 - y_1\)) is divided by the change in x (\(x_2 - x_1\)). This straightforward calculation helps us determine the direction and rate of change of the function. In the given exercise, using points (\((4, 7)\)) and (\((6, 11)\)), we calculated \(m = 2\). This means that for every one unit increase in x, y increases by two units in our line.

Point-Slope Form

The point-slope form of a linear equation provides an easy way to write the equation of a line when we know one point on that line and its slope. The general formula is:

• \( y - y_1 = m(x - x_1) \)

Here \((x_1,y_1)\) are the coordinates of a known point, and \(m\) is the slope. Using this form, any linear equation can be quickly established. From our example, with a slope of \(2\) and the point \((4, 7)\), the equation begins as:

• \( y - 7 = 2(x - 4) \)

This equation defines the line perfectly with every parameter filled in. It acts as the first step towards simplifying to a more recognizable form.

Equation of a Line

Once the point-slope equation is formed, our next goal is usually a y-intercept form (\(y = mx + b\)), which is popular because it's straightforward and easy to interpret. It expresses y directly as a function of x, clearly showing the slope and y-intercept. By simplifying the point-slope form

• \( y - 7 = 2(x - 4) \)

to get

• \( y = 2x - 1 \)

we now have expressed the equation in a more familiar way, where \(b = -1\), identifying the point where the line crosses the y-axis. Finally, substituting \(x = 5\) confirmed that \(f(5) = 9\). This verification assures that our function prediction aligns with the overall linear relationship.