Problem 88 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of the equation by hand. Verify using a graphing utility. $$y=\frac{3 x-5}{2}+2$$

Step-by-Step Solution

Answer

The graph of the equation \(y=\frac{3x-5}{2}+2\) is a line with slope 3/2 and y-intercept -1/2, passing through the points (0, -1/2) and (2, 2.5).

Step 1: Identify slope and y-intercept

The equation given is in the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. From the equation \(y=\frac{3x-5}{2}+2\), we see that the slope \(m\) = 3/2 and the y-intercept \(c\) = \(2 - 5/2 = -1/2\).

Step 2: Draw the y-intercept

The y-intercept is the point where the line crosses the y axis. This happens when the value of the x-coordinate is zero. In this case the y-intercept is -1/2, so the point to graph on the y-axis is (0, -1/2).

Step 3: Use the slope to find another point

The slope 3/2 means that for each 2 units increase in x, y increases by 3 units. Since we already have the point (0, -1/2) from the y-intercept, we can use the slope to find another point. Moving 2 units to the right (x = 2), y increases by 3 units, so the new y-coordinate will be \( -1/2 + 3 = 2.5 \). So, we have another point (2, 2.5).

Step 4: Draw the line

Now that we have two points, (0, -1/2) and (2, 2.5), we can draw a line passing through these points. This line represents the graph of the equation \(y=\frac{3x-5}{2}+2\).

Step 5: Verify the graph

To verify the graph, use a graphing utility and input the equation \(y=\frac{3x-5}{2}+2\). The line drawn on the graphing utility should match the line graphed by hand.

Key Concepts

Slope and Y-intercept

Understanding the slope and y-intercept of a linear equation is fundamental to graphing it accurately. A linear equation in the form of \( y = mx + c \) easily shows the slope \( m \) and the y-intercept \( c \).
In our exercise, the equation was given as \( y = \frac{3x-5}{2} + 2 \). Here's how we identify the components:

• The slope \( m \) tells you how steep the line is. Here the slope is \( \frac{3}{2} \), meaning for every 2 units we move to the right on the x-axis, we move up 3 units on the y-axis.
• The y-intercept \( c \) indicates where the line crosses the y-axis. We found it as \( -\frac{1}{2} \). This is the point (0, -0.5) where the line meets the y-axis.

Recognizing these parts allows us to quickly sketch and understand the behavior of the line on a graph.

Sketching Graphs

Once you know the slope and y-intercept, you're ready to sketch your graph. Start by plotting the y-intercept on the y-axis. In our exercise, that point was \( (0, -\frac{1}{2}) \).
The next step is using the slope to find another point. Since the slope was \( \frac{3}{2} \), this means you go "up 3, right 2." Starting from \( (0, -\frac{1}{2}) \), move right 2 units to \( x = 2 \) and up 3 units to \( y = 2.5 \). This gives us the point (2, 2.5).
With these two points, you can draw a straight line through them. This line is the visual representation of the equation and illustrates its path across the graph.

Verification Using Graphing Utilities

After sketching the graph by hand, it's good practice to verify your solution with a graphing utility.
Graphing utilities are powerful tools that help in checking the accuracy of your graph. You simply enter the equation, in this case, \( y = \frac{3x-5}{2} + 2 \), into the utility.
Observe the graph generated by the utility; it should match the line you drew manually through the points \( (0, -\frac{1}{2}) \) and (2, 2.5).

• Ensure that the line passes through the correct y-intercept.
• Check that the direction (upward or downward slope) is consistent.

Using graphing utilities not only validates your work but also enhances understanding by allowing you to visualize how changes in equations affect the graph.