Problem 92 - Algebra A Combined Function

Question

Find the slope of the line through the given points. The graph of \(y=-\frac{1}{3} x+2\) has a slope of \(-\frac{1}{3}\). The graph of \(y=-2 x+2\) has a slope of -2 . The graph of \(y=-4 x+2\) has a slope of -4 . Graph all three equations on a single coordinate system. As the absolute value of the slope becomes larger, how does the steepness of the line change?

Step-by-Step Solution

Answer

As the absolute value of the slope increases, the steepness of the line increases.

Step 1: Identify the Slope of Each Line

Each line is represented by the equation \[ y = mx + b \] where \( m \) is the slope. The given equations are: 1. \( y = -\frac{1}{3}x + 2 \) with a slope of \( -\frac{1}{3} \).2. \( y = -2x + 2 \) with a slope of \( -2 \). 3. \( y = -4x + 2 \) with a slope of \( -4 \). Thus, the slopes are \( -\frac{1}{3}, -2, \) and \( -4 \) respectively.

Step 2: Understand the Relationship Between Slope and Steepness

The slope \( m \) indicates how steep the line is. A larger absolute value of \( m \) means a steeper line. Comparing the slopes,- \( -\frac{1}{3} \) is the least steep,- \( -2 \) is steeper than \( -\frac{1}{3} \),- \( -4 \) is the steepest line among the three.

Step 3: Graph Each Equation

Plot each line on the same set of axes. Start by plotting the y-intercept (\( b = 2 \) for all lines) at the point (0,2).- For \( y = -\frac{1}{3}x + 2 \), move down 1 unit and right 3 units from the y-intercept for the next point.- For \( y = -2x + 2 \), move down 2 units and right 1 unit from the y-intercept for the next point.- For \( y = -4x + 2 \), move down 4 units and right 1 unit from the y-intercept for the next point. Draw each line through these points.

Key Concepts

Linear Equations

Linear equations are fundamental in understanding the relationship between variables. They are usually written in the form \[ y = mx + b \] where:

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

In linear equations, the value of \( m \) determines how the line slants as you move along the x-axis. If \( m \) is positive, the line goes up; if it's negative, the line goes down. The y-intercept, \( b \), helps you identify the starting point on the y-axis. Whenever you encounter a linear equation, break it down by first finding \( m \) and \( b \). This makes it easier to graph the line and understand its behavior.

Graphing a Line

Graphing a line from a linear equation involves understanding its slope and y-intercept. To graph the equation \[ y = mx + b \], follow these steps:

• Start by plotting the y-intercept \( b \) on the y-axis. This is the point \((0, b)\).
• Use the slope \( m \) to determine the direction of the line. The slope \( m \) is often written as a fraction: \( \frac{rise}{run} \).
• From the y-intercept, move vertically by the "rise" value and horizontally by the "run" value.
• Mark this new point and draw a line through the y-intercept and this point.

For example, in the equation \( y = -2x + 2 \), start at the y-intercept (0,2). From there, the slope \(-2\) tells you to move down 2 units and right 1 unit to find your next point. Connect these points with a straight line and continue it across the grid to complete your graph.

Linear Functions

Linear functions are a type of function that produce straight lines when graphed. They are crucial in math because they form the basis for understanding more complex functions. Here's why linear functions are important:

• They provide a simple model to show constant rates of change, which means for every consistent change in \( x \), \( y \) changes by a fixed amount, determined by the slope.
• They can be used to predict values outside the initial data set within a reasonable range.
• Understanding linear functions helps in financial projections, physics problems, and any scenario involving uniform rates.

For example, the equation \( y = -\frac{1}{3}x + 2 \) describes a linear function where for every 3 units increase in \( x \), \( y \) decreases by 1 unit. It represents a constant, predictable relationship between \( x \) and \( y \). By mastering these functions, you gain a powerful tool for mathematical modeling.