Problem 33 - College Algebra with Modeling and Visualization

Question

Exercises \(33-38:\) Write a formula for a linear function f whose graph satisfies the conditions. $$ \text { Slope }-\frac{3}{4}, y \text { -intercept } \frac{1}{3} $$

Step-by-Step Solution

Answer

The formula is \( f(x) = -\frac{3}{4}x + \frac{1}{3} \).

Step 1: Understand the Linear Function Formula

A linear function can be represented in the form \( f(x) = mx + b \), where \( m \) is the slope, and \( b \) is the \( y \)-intercept. We need to identify these values from the given conditions.

Step 2: Substitute the Given Slope

Substitute the slope \( m = -\frac{3}{4} \) into the linear function formula: \( f(x) = -\frac{3}{4}x + b \).

Step 3: Substitute the Given Y-intercept

Substitute the \( y \)-intercept \( b = \frac{1}{3} \) into the equation from Step 2: \( f(x) = -\frac{3}{4}x + \frac{1}{3} \).

Step 4: Write the Final Formula

The formula for the linear function that satisfies the given conditions is \( f(x) = -\frac{3}{4}x + \frac{1}{3} \).

Key Concepts

Slope of a Line

The slope of a line is a crucial concept within linear functions and geometry. The slope, often represented as \( m \), essentially measures the steepness or incline of a line. You can think of it as the amount of vertical change for every unit of horizontal change. Mathematically, the slope is calculated by taking the difference in \( y \)-values and dividing it by the difference in \( x \)-values between two points on the line:
\[ m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \]

In our example, the given slope is \(-\frac{3}{4}\). This means for every 4 units you move horizontally to the right, the line moves 3 units vertically down, indicating a decreasing or negative slope. The slope direction determines whether the line moves upward or downward as it progresses from left to right on the graph. A positive slope means the line rises, while a negative slope, like \(-\frac{3}{4}\), means it falls.

• Positive slope: Line increases from left to right
• Negative slope: Line decreases from left to right
• Zero slope: Horizontal line
• Undefined slope: Vertical line

Y-intercept

The \( y \)-intercept is another fundamental concept in understanding linear functions. Represented by \( b \) in the linear equation \( f(x) = mx + b \), the \( y \)-intercept is the point where the line crosses the \( y \)-axis. When graphing, it provides the starting point on the \( y \) axis, from which you apply the slope to determine the direction and angle of the line.

In the context of this problem, the \( y \)-intercept is \( \frac{1}{3} \). This means that when \( x = 0 \), \( f(x) \), or \( y \), equals \( \frac{1}{3} \). If you imagine drawing the line on a graph, you'd start by plotting the point \( (0, \frac{1}{3}) \) and from this point, you follow the slope to draw the rest of the line. The beauty of the \( y \)-intercept is how it provides clarity in positioning where the line begins on the vertical axis, distinct from the slope's guidance in line direction.

• \( y \)-intercept: The \( y \)-value where \( x \) is 0
• Affects the line's vertical position on a graph

Linear Equations

Linear equations are foundational to understanding mathematical relationships and represent relationships between variables. The standard form of a linear function is \( f(x) = mx + b \), where \( m \) and \( b \) hold specific roles:
- \( m \) represents the slope
- \( b \) represents the \( y \)-intercept

Linear equations model straight-line relationships where the value of \( y \) changes at a consistent rate with respect to \( x \). This consistency is visually represented by straight lines when plotted on a Cartesian plane.

For the equation \( f(x) = -\frac{3}{4}x + \frac{1}{3} \) derived from the exercise:

• The slope \(-\frac{3}{4}\) dictates a downward angle
• The \( y \)-intercept \(\frac{1}{3}\) provides the initial position on the \( y \) axis

This format helps in predicting and understanding how changes in \( x \) affect \( y \), making linear equations a powerful tool in diverse fields like physics, economics, and beyond. They effectively describe systems where the relationship between variables is additive and consistent across the board.