Problem 69
Question
Consider the equation \(c=3 n+2\) (a) Sketch the graph of this equation using the horizontal axis for the \(n\) values and the vertical axis for the \(c\) values. (b) Use the graph to describe what happens to the values of \(c\) as \(n\) changes. (c) Sketch the graph of this equation using the horizontal axis for the \(c\) values and the vertical axis for the \(n\) values. (d) Use the graph to describe what happens to the values \(n\) as \(c\) changes. (e) Does the way we choose to label the axes affect the appearance of the graph? Does the way we choose to label the axes affect the relationship between the variables?
Step-by-Step Solution
Verified Answer
Swapping the axes changes the graph's appearance but not the relationship. As \(n\) increases, \(c\) increases linearly and vice versa.
1Step 1: Rewrite the Equation
Rewrite the given equation in standard form: \[ c = 3n + 2 \]
2Step 2: Sketch the Graph with n on Horizontal Axis
Plot the graph of the equation with \( n \) on the horizontal axis and \( c \) on the vertical axis.1. Identify intercepts. The y-intercept (value of \( c \) when \( n \) is 0) is 2.2. Plot this point: (0,2).3. Determine another point by choosing any value for \( n \) (e.g., \( n = 1 \)), then find \( c = 3(1) + 2 = 5 \).4. Plot this point: (1,5).5. Draw a straight line through these points to complete the graph.
3Step 3: Describe Changes in c as n Changes
As \( n \) increases, the value of \( c \) increases linearly. Specifically, for every increase of 1 in \( n \), \( c \) increases by 3.
4Step 4: Sketch the Graph with c on Horizontal Axis
Plot the graph of the equation with \( c \) on the horizontal axis and \( n \) on the vertical axis.1. Rearrange the equation to solve for \( n \): \( n = \frac{c - 2}{3} \).2. Identify intercepts. The x-intercept (value of \( c \) when \( n \) is 0) is 2.3. Plot this point: (2,0).4. Determine another point by choosing any value for \( c \) (e.g., \( c = 5 \)), then find \( n = \frac{5 - 2}{3} = 1 \).5. Plot this point: (5,1).6. Draw a straight line through these points to complete the graph.
5Step 5: Describe Changes in n as c Changes
As \( c \) increases, the value of \( n \) increases linearly. Specifically, for every increase of 3 in \( c \), \( n \) increases by 1.
6Step 6: Discuss the Labeling of Axes
The graph's appearance does change when the axes are swapped, transforming the slope to reflect the inverse relationship. However, the original relationship between the variables \(c\) and \(n\) remains intact irrespective of their positions on the axes.
Key Concepts
graphing linear equationsslope and interceptvariable relationship
graphing linear equations
Graphing linear equations helps us visualize the relationship between two variables. Consider the equation \[ c = 3n + 2 \], where 'n' is the independent variable and 'c' is the dependent variable.
To graph this equation, we start by identifying key points. First, the y-intercept, where the line crosses the vertical axis, is 2 because when 'n' is 0, \[ c = 3 \times 0 + 2 = 2 \]. Mark this point on the graph at (0,2).
Next, choose another 'n' value, like 1. Substitute it into the equation: \[ c = 3 \times 1 + 2 = 5 \]. Plot this point at (1,5). Connect these points with a straight line.
For graphing the same equation with swapped axes, solve for 'n': \[ n = \frac{c - 2 }{3} \]. The x-intercept (value of 'c' when 'n' is 0) is 2. Plot it at (2,0). Choose another 'c' value, like 5: \[ n = \frac{5 - 2}{3} = 1 \]. Plot the point (5,1) and connect the points with a straight line.
To graph this equation, we start by identifying key points. First, the y-intercept, where the line crosses the vertical axis, is 2 because when 'n' is 0, \[ c = 3 \times 0 + 2 = 2 \]. Mark this point on the graph at (0,2).
Next, choose another 'n' value, like 1. Substitute it into the equation: \[ c = 3 \times 1 + 2 = 5 \]. Plot this point at (1,5). Connect these points with a straight line.
For graphing the same equation with swapped axes, solve for 'n': \[ n = \frac{c - 2 }{3} \]. The x-intercept (value of 'c' when 'n' is 0) is 2. Plot it at (2,0). Choose another 'c' value, like 5: \[ n = \frac{5 - 2}{3} = 1 \]. Plot the point (5,1) and connect the points with a straight line.
slope and intercept
The slope and intercept are crucial elements in understanding linear equations. In the equation \[ c = 3n + 2 \], the slope is 3, and the y-intercept is 2.
The slope provides information about how steep the line is and the direction it goes. A slope of 3 means that for every 1 unit increase in 'n', 'c' increases by 3. It's the 'rise over run'.
The y-intercept is the point where the line crosses the y-axis. Here, it's 2, showing that when 'n' is 0, 'c' is 2.
When we switch the axes (\[ n = \frac{c - 2}{3} \]), the roles of slope and intercept change but the relationship remains the same. Now, the slope is \[ \frac{1}{3} \], and the x-intercept is 2. This tells us that for every increase of 3 in 'c', 'n' increases by 1.
The slope provides information about how steep the line is and the direction it goes. A slope of 3 means that for every 1 unit increase in 'n', 'c' increases by 3. It's the 'rise over run'.
The y-intercept is the point where the line crosses the y-axis. Here, it's 2, showing that when 'n' is 0, 'c' is 2.
When we switch the axes (\[ n = \frac{c - 2}{3} \]), the roles of slope and intercept change but the relationship remains the same. Now, the slope is \[ \frac{1}{3} \], and the x-intercept is 2. This tells us that for every increase of 3 in 'c', 'n' increases by 1.
variable relationship
Understanding the relationship between variables in an equation is key in many areas. In \[ c = 3n + 2 \], we see a linear relationship between 'n' and 'c'.
When 'n' increases by 1, 'c' increases by 3. This consistency is what makes the relationship linear. It’s predictable, making it easier to graph and analyze.
The inverse is seen when solving for 'n': \[ n = \frac{c - 2}{3} \]. Here, we find that for every increase of 3 in 'c', 'n' increases by 1, maintaining a steady ratio.
This understanding helps answer questions about changes. For example, as 'n' doubles, 'c' also changes predictably, providing clear insights into how the variables affect each other.
When 'n' increases by 1, 'c' increases by 3. This consistency is what makes the relationship linear. It’s predictable, making it easier to graph and analyze.
The inverse is seen when solving for 'n': \[ n = \frac{c - 2}{3} \]. Here, we find that for every increase of 3 in 'c', 'n' increases by 1, maintaining a steady ratio.
This understanding helps answer questions about changes. For example, as 'n' doubles, 'c' also changes predictably, providing clear insights into how the variables affect each other.
Other exercises in this chapter
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