Problem 26 - College Algebra with Modeling and Visualization

Question

State the slope of the graph of \(f\). Interpret this slope. $$ f(x)=\frac{2}{3} x $$

Step-by-Step Solution

Answer

The slope is \( \frac{2}{3} \), indicating the graph rises slowly.

Step 1: Identify the Function Type

The function given is in the form of a linear equation, which generally takes the form of \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Extract the Slope

The given equation is \( f(x) = \frac{2}{3}x \). Here, \( \frac{2}{3} \) is the coefficient of \( x \), which represents the slope \( m \) of the line.

Step 3: Interpret the Slope

The slope of \( \frac{2}{3} \) means that for every unit increase in \( x \), the value of \( f(x) \) increases by \( \frac{2}{3} \). This describes the steepness and direction of the line, indicating that the line rises gently from left to right.

Key Concepts

Linear equations

A linear equation is one where each term is either a constant or the product of a constant and the variable. Its most familiar form is the slope-intercept form, represented as \( y = mx + b \). Here, \( m \) stands for the slope of the line and \( b \) is the y-intercept. The y-intercept is where the line crosses the y-axis. In the equation of the function \( f(x) = \frac{2}{3}x \), it appears in its simplest form. Here, the y-intercept \( b \) is 0 because there is no constant term added to the equation. This tells us that the line passes through the origin (0,0) on the graph. Linear equations are called so because they produce straight line graphs when plotted. These lines have consistent slopes, which defines their constant rate of change.
Understanding the basic structure of linear equations is essential as it helps distinguish other more complex functions you'll encounter. By mastering the concepts of slopes and y-intercepts, analyzing more complex equations becomes significantly easier.

Coefficient interpretation

The coefficient in a linear equation, represented by \( m \) in \( y = mx + b \), is crucial because it dictates the slope of the line. The slope is the ratio of the change in \( y \) to the change in \( x \), often referred to as "rise over run." In \( f(x) = \frac{2}{3}x \), the coefficient of \( x \) is \( \frac{2}{3} \). This means that for each increase of 1 in \( x \), the \( y \) value (or \( f(x) \)) increases by \( \frac{2}{3} \).
Interpreting coefficients involves:

• Recognizing it as the slope, which gives directionality to the line.
• Understanding the sign of the coefficient: positive indicates a line rising from left to right, while negative would point downward.
• Linking steeper coefficients with larger numbers: bigger numerators or smaller denominators indicate a steeper line.

It is important to always interpret the coefficient in context, as it truly informs us about how a function behaves and changes.

Graph interpretation

Graph interpretation involves visualizing and analyzing the characteristics of a graphed function. When you plot the function \( f(x) = \frac{2}{3}x \), the resulting graph is a straight line that passes through the origin. This is because the equation in slope-intercept form lacks a constant (\( b = 0 \)), denoting it doesn't shift up or down along the y-axis.
Understanding the graph:

• The slope is \( \frac{2}{3} \), indicating that the line rises gently to the right. Visually, this means moving 3 units to the right along the x-axis increases the y-value by 2 units.
• The line's intersection with the y-axis at (0,0) shows no initial value when \( x = 0 \).
• Orientation and slope show how the line behaves over intervals — predicting value increases over specific x changes.

A graph provides an immediate image of the rate at which values change, offering insights into solutions and real-world applications. Seeing equations graphically enhances understanding and offers another way to verify calculated values.