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Magazine Chapter 3: Problem 12
In Exercises \(5-36,\) graph the given functions. $$A=6-\frac{1}{3} r$$
Short Answer
Expert verified
The graph of \( A = 6 - \frac{1}{3} r \) is a straight line with a y-intercept at 6 and a slope of \(-\frac{1}{3}\).
Step by step solution
01
Identify the Equation Type
The equation given is a linear equation: \( A = 6 - \frac{1}{3} r \). This represents a linear function, which is typically graphed as a straight line. Let's identify the slope and the y-intercept to help us graph.
02
Determine the Slope and Y-Intercept
The equation is in the slope-intercept form, \( y = mx + b \), where \(m\) represents the slope and \(b\) is the y-intercept. Comparing \( A = 6 - \frac{1}{3} r \) to \( y = mx + b \), we see that the slope \(m\) is \(-\frac{1}{3}\) and the y-intercept \(b\) is 6.
03
Plot the Y-Intercept
To start graphing, plot the y-intercept on the graph at \( (0, 6) \). This is the point where the graph crosses the y-axis.
04
Use the Slope to Find Another Point
The slope \(-\frac{1}{3}\) indicates that for every 1 unit increase in \(r\) (the horizontal axis), the value of \(A\) decreases by \(\frac{1}{3}\). From the y-intercept \( (0, 6) \), move 1 unit to the right to \((1, ? )\). Since the slope is negative \(-\frac{1}{3}\), go down \(\frac{1}{3}\) units to arrive at \((1, 6 - \frac{1}{3})\).
05
Plot Additional Points
Continuing from the point \((1, 5.67)\), repeat using the slope \(-\frac{1}{3}\) to find another point. Move from \((1, 5.67)\) to \((2, 5.33)\) by moving 1 unit right and \(\frac{1}{3}\) unit down. Keep following this pattern for more points.
06
Draw the Graph
Plot the identified points on graph paper or a digital tool. Draw a line through them; this line represents the graph of the equation \( A = 6 - \frac{1}{3} r \). Ensure the line extends across the graph, keeping in mind the slope and intercept.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope-Intercept Form
The slope-intercept form is a way of expressing linear equations in the form \( y = mx + b \). This format is especially useful for quickly identifying important features of the graph of a line, including the slope and the y-intercept. In this form:
- \( m \) represents the slope of the line.
- \( b \) indicates the y-intercept, where the line will intersect the y-axis.
Graphing Lines
When graphing lines from an equation, understanding the slope-intercept form can greatly simplify the process. Start by focusing on:
- The y-intercept: Plot this point first as it's the starting point of your line on the y-axis.
- The slope: This indicates the direction and steepness of the line.
Y-Intercept
The y-intercept is the point where a line crosses the y-axis in a graph. For linear equations in slope-intercept form, it appears as the constant \( b \). In our example \( A = 6 - \frac{1}{3} r \), the y-intercept is 6. This means that when \( r \) is zero, \( A \) equals 6.Plotting the y-intercept is your first step in graphing. Simply find the y-coordinate and mark it on the y-axis. It's crucial because it serves as the reference point for drawing the entire line, providing a starting position for using the slope.
Slope
The slope of a line indicates its steepness and direction. It tells us how much the graph rises or falls as it moves from left to right. In the slope-intercept form \( y = mx + b \), \( m \) is the slope.A positive slope means the line moves upwards, whereas a negative slope indicates a downward direction. For the equation \( A = 6 - \frac{1}{3} r \), the slope is \(-\frac{1}{3}\). This tells us that for every one unit increase in \( r \), \( A \) decreases by \( \frac{1}{3} \) units. Understanding the slope helps in plotting additional points on the graph and ensures the accuracy of the graph's representation.
