Which Graph Has A Rate Of Change Equal To 1/3

Which Graph Has a Rate of Change Equal to 1/3? Understanding Slope and Linear Equations

Determining which graph possesses a rate of change equal to 1/3 requires a fundamental understanding of slope and linear equations. The rate of change, in this context, refers to the slope of a line on a graph. A slope of 1/3 indicates that for every 3 units of horizontal movement (change in x), there's a corresponding 1-unit vertical movement (change in y). This article will delve into various aspects of identifying such graphs, exploring different representations of linear relationships, and providing practical examples to solidify your comprehension.

Understanding Slope and its Significance

The slope of a line is a crucial concept in algebra and calculus. It quantifies the steepness and direction of a line. A positive slope indicates an upward trend (from left to right), while a negative slope indicates a downward trend. A slope of zero represents a horizontal line, and an undefined slope represents a vertical line.

The formula for calculating the slope (m) of a line given two points (x1, y1) and (x2, y2) is:

m = (y2 - y1) / (x2 - x1)

In our case, we're looking for a graph with a slope of 1/3. This means the change in y is one-third the change in x.

Identifying Graphs with a Slope of 1/3

Several representations can depict a linear relationship with a slope of 1/3:

1. Graphical Representation

Visually identifying a graph with a slope of 1/3 involves careful observation. Locate two distinct points on the line. Then, calculate the vertical change (rise) and the horizontal change (run) between these points. The ratio of rise to run should equal 1/3.

For example, if you identify points (3, 1) and (6, 2) on a line, the slope is calculated as:

m = (2 - 1) / (6 - 3) = 1/3

Key Visual Cue: The line will have a gentle, positive incline. It won't be as steep as a line with a slope of 1 or 2, nor as shallow as a line with a slope of 1/10.

2. Equation of a Line

The equation of a line is typically expressed in slope-intercept form:

y = mx + b

Where:

• 'm' represents the slope
• 'b' represents the y-intercept (the point where the line crosses the y-axis)

To identify a graph with a slope of 1/3 from its equation, simply check the coefficient of 'x'. If it's 1/3, then the graph represents a line with the desired slope.

Example: The equation y = (1/3)x + 2 represents a line with a slope of 1/3 and a y-intercept of 2.

3. Table of Values

A table of values displays corresponding x and y coordinates of points on a line. You can determine the slope by selecting any two pairs of coordinates from the table and applying the slope formula. If the calculated slope consistently equals 1/3 for different pairs, the table represents a line with the desired slope.

Example:

Using points (3, 1) and (6, 2): m = (2 - 1) / (6 - 3) = 1/3

Using points (0, 0) and (9, 3): m = (3 - 0) / (9 - 0) = 1/3

4. Real-World Applications

Many real-world scenarios can be modeled using linear equations with a slope of 1/3. For instance, consider a scenario where a water tank fills at a rate of 1 gallon every 3 minutes. The rate of change (gallons per minute) is 1/3. A graph depicting the water level over time would have a slope of 1/3. Similarly, a car traveling at a speed of 20 miles every 60 minutes (1 hour) has a speed (miles per minute) with a slope of 1/3.

Distinguishing Between Different Slopes

It's crucial to differentiate between a slope of 1/3 and other slopes.

• Steeper Slopes (e.g., 1, 2, 3): These lines will have a more pronounced incline than a line with a slope of 1/3.
• Shallower Slopes (e.g., 1/10, 1/5): These lines will be flatter than a line with a slope of 1/3.
• Negative Slopes: These lines will have a downward incline from left to right.
• Zero Slope: This represents a horizontal line.
• Undefined Slope: This represents a vertical line.

Practical Exercises

Let's solidify your understanding with some practical exercises:

Exercise 1:

Determine if the following equations represent lines with a slope of 1/3:

a) y = (1/3)x - 5 b) y = 3x + 2 c) y = (1/3)x + 1/3 d) 2y = x + 6

Solutions:

a) Yes. The coefficient of x is 1/3. b) No. The coefficient of x is 3. c) Yes. The coefficient of x is 1/3. d) Yes. Rewriting the equation in slope-intercept form gives y = (1/2)x + 3. This is not 1/3.

Exercise 2:

Given the points (3, 2) and (6, 3), calculate the slope and determine if it equals 1/3.

Solution:

m = (3 - 2) / (6 - 3) = 1/3. Yes, the slope is 1/3.

Exercise 3:

Analyze the following table of values and determine if it represents a line with a slope of 1/3.

Solution:

Using points (0, 2) and (3, 3): m = (3 - 2) / (3 - 0) = 1/3. Using points (3, 3) and (6, 4): m = (4-3) / (6-3) = 1/3. Using points (6, 4) and (9, 5): m = (5-4)/(9-6) = 1/3. Yes, the table represents a line with a slope of 1/3.

Conclusion

Identifying a graph with a rate of change equal to 1/3 involves understanding the concept of slope and its various representations. By using the slope formula, analyzing equations, examining tables of values, and observing visual cues on the graph, you can confidently determine which graphs exhibit this specific rate of change. This understanding is fundamental in various mathematical and real-world applications. Remember to practice these methods to build a strong grasp of linear relationships and their graphical representations.