Problem 34 - College Algebra with Modeling and Visualization

Question

Exercises \(33-38:\) Write a formula for a linear function f whose graph satisfies the conditions. Slope \(-122, y\) -intercept 805

Step-by-Step Solution

Answer

The formula is \( f(x) = -122x + 805 \).

Step 1: Understand Linear Function Formula

The general formula for a linear function is \( f(x) = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept.

Step 2: Substitute the Given Slope

Substitute the given slope, \( m = -122 \), into the formula replacing \( m \). The function becomes \( f(x) = -122x + b \).

Step 3: Substitute the Given Y-Intercept

Substitute the given y-intercept, \( b = 805 \), into the formula replacing \( b \). The function becomes \( f(x) = -122x + 805 \).

Step 4: Write the Final Function Formula

Now that both the slope and the y-intercept have been substituted, the final formula for the linear function is \( f(x) = -122x + 805 \).

Key Concepts

Slope

The slope of a line is a key concept in understanding linear equations. In simple terms, the slope depicts how steep a line is. It measures the rate of change in the vertical direction relative to the horizontal direction. The slope is represented by the letter \(m\) in the equation of a line. It shows how much \(y\) changes when \(x\) changes by one unit.

• A positive slope means the line is rising as it moves from left to right.
• A negative slope indicates that the line is falling as it moves from left to right.
• A slope of zero implies a horizontal line which means no change in \(y\) with changes in \(x\).
• An undefined slope indicates a vertical line, where \(x\) does not change, but \(y\) does.

Understanding the slope helps predict how two quantities are related. In the given exercise, the slope is \(-122\), meaning for every unit increase in \(x\), \(y\) decreases by 122 units, signifying a downward trend.

y-intercept

The y-intercept is another crucial aspect of linear functions. It tells us the point where the line crosses the y-axis. This is important because it provides a starting value when performing calculations. The y-intercept is represented by the letter \(b\) in the linear function equation.

• When a line crosses the y-axis, it is at the point where \(x\) equals zero.
• The y-intercept gives the value of \(f(x)\) when \(x = 0\).
• It is a constant in the linear equation, meaning it does not change for different values of \(x\).

In the provided example, the y-intercept is 805, meaning at the point where the line crosses the y-axis, \(y\) equals 805.
Knowing the y-intercept allows you to construct the graph of the linear function more efficiently.

Linear Equations

Linear equations form the foundation of understanding relationships between two variables. They are equations of the first order, meaning they have no exponents greater than one. Linear equations are often written in the slope-intercept form: \(f(x) = mx + b\).

• \(m\) represents the slope, showing the steepness/direction of the line.
• \(b\) represents the y-intercept, showing where the line crosses the y-axis.
• It forms a straight line when graphed.
• It is useful in predicting values, modeling real-world scenarios, and analyzing trends.

By understanding linear equations, many real-world problems can be translated into a simple mathematical form which is easy to analyze. In this exercise, using the slope \(-122\) and y-intercept 805, the linear equation can be formulated as \(f(x) = -122x + 805\), clearly illustrating the relationship between the dependent and independent variables.