Problem 35 - College Algebra with Modeling and Visualization

Question

Exercises \(33-38:\) Write a formula for a linear function f whose graph satisfies the conditions. Slope \(15,\) passing through the origin

Step-by-Step Solution

Answer

The formula is \( f(x) = 15x \).

Step 1: Identify the Form of the Equation

For a linear function, we use the point-slope form equation which is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Substitute the Given Slope

The problem states the slope \( m = 15 \). Substitute this value into the equation to get \( y = 15x + b \).

Step 3: Identify the Y-Intercept

Since the line passes through the origin, the point \((0,0)\) is on the line, meaning the y-intercept \( b = 0 \).

Step 4: Write the Final Equation of the Linear Function

Substituting 0 for \( b \) in the equation gives \( y = 15x \). This is the formula for the linear function.

Key Concepts

Understanding Slope

The slope of a line is a crucial concept in understanding linear functions. It is represented by the letter \( m \) in the equation of a line, specifically in forms like the slope-intercept form \( y = mx + b \). The slope tells us how steep the line is and the direction it is trending. There are a few important things to remember about slope:

• If the slope is positive, the line goes upwards from left to right.
• If the slope is negative, the line slopes downwards from left to right.
• A larger absolute value of the slope means a steeper line.
• A slope of 0 indicates a horizontal line, which means no change in y as x changes.

In the context of the problem, the slope of 15 tells us that for every 1 unit increase in x, the value of y increases by 15 units, resulting in a quite steep line that rises sharply.

The Role of the Y-Intercept

The y-intercept is the point where the line crosses the y-axis on a graph. In the equation \( y = mx + b \), \( b \) represents the y-intercept. It is the value of y when x is 0, providing a starting point to plot the line on a graph. A few key points about the y-intercept:

• It is essential for shifting the line up or down on the graph without changing its slope.
• If \( b = 0 \), as in the problem, the line passes through the origin, which is the point \( (0,0) \).
• The sign of the y-intercept will tell us if the line is above or below the origin when it intersects the y-axis.

In this problem, since the line passes through the origin, the y-intercept \( b \) is 0. This signifies that the linear graph starts right at the origin and climbs steeply upwards with each increase in x.

Point-Slope Form Equation

The point-slope form equation is one of several ways to express the equation of a line. It is particularly handy when you know a point through which the line passes and the line's slope. The typical form of this equation is \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a point on the line, and \( m \) is the slope. Here is why the point-slope form is useful:

• It simplifies creating a line equation if you do not initially have the y-intercept.
• You can quickly substitute any known point and the slope into the formula to describe the full line.
• It helps in problems where you're given a specific point outside the origin.

For the given exercise, knowing that the line passes through the origin and has a slope of 15, the equation simplifies directly to \( y = 15x \). This transformation showcases how knowledge of specific line characteristics leads directly to the line's equation by simplifying point-slope to slope-intercept form.