Problem 36 - College Algebra with Modeling and Visualization

Question

Exercises \(33-38:\) Write a formula for a linear function f whose graph satisfies the conditions. Slope 1.68 , passing through \((0,1.23)\)

Step-by-Step Solution

Answer

The formula is \( f(x) = 1.68x + 1.23 \).

Step 1: Identify the Components of the Linear Function

The general formula for a linear function is \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The slope is given as 1.68, and the point \((0, 1.23)\) indicates the y-intercept.

Step 2: Substitute Known Values into the Formula

Using the slope \( m = 1.68 \) and the point \((0, 1.23)\), which tells us that the y-intercept \( b = 1.23 \), substitute these into the linear equation: \( f(x) = 1.68x + 1.23 \).

Step 3: Verify the Equation

To ensure the formula is correct, check if the graph of the function passes through \((0, 1.23)\). When \( x = 0 \), \( f(0) = 1.68(0) + 1.23 = 1.23 \), confirming the formula's accuracy.

Key Concepts

Slope

The slope of a linear function is a key concept that describes how steep a line is. This is symbolized by the letter 'm' in the linear equation format, which is typically written as \(f(x) = mx + b\). A positive slope, like in our case where \(m = 1.68\), indicates that as the x-value increases, the y-value also rises.
The slope is calculated by taking the ratio of the change in the vertical direction (y-axis) to the change in the horizontal direction (x-axis). This can often be expressed as:

• Slope \(m\) = \(\frac{\text{Change in y}}{\text{Change in x}}\)

In simple terms, think of the slope as a way to measure the angle of the line. A larger positive slope means a steeper line, while a smaller one indicates a gentler incline.

Y-intercept

The y-intercept is where the line crosses the y-axis on a graph. It's represented by the letter 'b' in the linear equation \(f(x) = mx + b\). This point occurs when \(x = 0\), which means the y-intercept is essentially the output value of the function when there is no input from x.
In the equation provided, \(b = 1.23\), which means the line crosses the y-axis at 1.23. It tells us the starting value of the function before any changes in x are applied.
Understanding the y-intercept helps in easily drawing the graph of the function because it's one of the two critical points needed to define a straight line.

Graphing Linear Equations

Graphing linear equations involves plotting the important points determined by the slope and y-intercept onto a grid. With the equation \(f(x) = 1.68x + 1.23\), the slope \(m = 1.68\) and y-intercept \(b = 1.23\) guide this process.

• Start by plotting the y-intercept. In this scenario, plot a point at (0, 1.23) on the y-axis.
• Next, use the slope to determine the direction and steepness of the line. Since \(m = 1.68\), move 1 unit to the right and 1.68 units up to plot another point.
• Draw a straight line through these plotted points to extend the graph on both sides.

Remember, the linear equation represents a straight line, so only two points are needed to graph it correctly. This visualization helps in understanding how the function behaves and how different values of \(x\) affect \(f(x)\).