Problem 23 - College Algebra with Modeling and Visualization

Question

State the slope of the graph of \(f\). Interpret this slope. $$ f(x)=2 x+7 $$

Step-by-Step Solution

Answer

The slope is 2, meaning the graph rises 2 units for each unit increase in \(x\).

Step 1: Identify the Equation of the Line

The provided function is in the form of a linear equation: \(f(x) = 2x + 7\). This equation represents a straight line.

Step 2: Determine the Slope

In the equation of a line \(y = mx + b\), \(m\) is the slope. For the function \(f(x) = 2x + 7\), the slope \(m\) is \(2\).

Step 3: Interpret the Slope

The slope of \(2\) indicates that for every unit increase in \(x\), the value of \(f(x)\) increases by \(2\). This means the line rises 2 units vertically for each 1 unit it moves horizontally.

Key Concepts

Linear Equations

Linear equations are fundamental in algebra and describe a straight line when you plot them on a graph. They take the form of \( y = mx + b \), where:

• \( y \) is the dependent variable.
• \( x \) is the independent variable.
• \( m \) represents the slope of the line.
• \( b \) is the y-intercept, indicating the point where the line crosses the y-axis.

The simplicity of linear equations makes them easy to solve and graph.
These equations are incredibly useful for modeling relationships between two variables where change is consistent. Think of the line as a path that shows how one variable responds to changes in the other.

Slope Interpretation

Understanding slope is crucial for interpreting linear equations. The slope, often represented by \( m \), tells us how steep a line is and its direction.
For the equation \( f(x) = 2x + 7 \), the slope is \( 2 \). This is a positive slope, indicating the line inclines upwards as it moves from left to right.**Slope Characteristics:**

• A positive slope means the function grows as \( x \) increases.
• A slope of \( 2 \) specifically implies that for each increase of \( 1 \) in \( x \), \( f(x) \) increases by \( 2 \).
• If the slope were negative, the line would descend.

The slope's numerical value measures how quickly changes occur. In practical terms, it might represent speed, cost per item, or other rates of change in everyday situations.

Graphing Functions

Graphing functions is a way to visually interpret the relationship between variables shown in equations. When graphing the function \( f(x) = 2x + 7 \), you start with its linear form and key components:

• The slope \( 2 \) guides the steepness and direction of the line.
• The y-intercept \( 7 \) shows where the line crosses the y-axis.

Steps for graphing:1. **Start at the y-intercept (\(0, 7\))**: Plot the point on the graph where the line intersects the y-axis.2. **Use the slope to find another point**: - From \( (0, 7) \), move 1 unit right along the x-axis. - Depending on the slope, move up (positive slope) or down (negative slope) on the y-axis. Here, move up \(2\) units to reach \((1, 9)\).3. **Draw the line through these points**: Extend it in both directions to complete the graph.Graphing helps interpret abstract equations as tangible visuals, making it easier to understand the function's behavior.