Problem 60 - Algebra for College Students

Question

Graph each equation on a graphing calculator using a window that shows both intercepts. Then use the appropriate feature of your calculator to find the intercepts. $$y=2-3 x$$

Step-by-Step Solution

Answer

Y-intercept: (0, 2); X-intercept: \(\left( \frac{2}{3}, 0 \right)\)

Step 1 - Set Up the Equation

Ensure the equation is in the form suitable for graphing. The given equation is already in the slope-intercept form: \(y = 2 - 3x\).

Step 2 - Enter the Equation in the Calculator

Turn on your graphing calculator and enter the equation \(y = 2 - 3x\) into the 'Y=' menu.

Step 3 - Adjust the Viewing Window

Set your viewing window to show both intercepts. A suitable window could be: \(X_{min} = -10\), \(X_{max} = 10\), \(Y_{min} = -10\), and \(Y_{max} = 10\).

Step 4 - Graph the Equation

Press 'Graph' to display the graph of the equation \(y = 2 - 3x\).

Step 5 - Find the Intercepts

Use the 'Calc' function on your graphing calculator to find the x-intercept (where \(y = 0\)) and y-intercept (where \(x = 0\)).

Step 6 - Record the Intercepts

For the y-intercept, set \(x = 0\): \(y = 2\). For the x-intercept, set \(y = 0\): solve the equation \(0 = 2 - 3x\) to get \(x = \frac{2}{3}\). Therefore, the y-intercept is (0, 2) and the x-intercept is \(\left( \frac{2}{3}, 0 \right)\).

Key Concepts

Understanding Slope-Intercept Form

The slope-intercept form is a way to write the equation of a straight line. It's given as:

\[ y = mx + b \]

Here:

• m - represents the slope of the line, which indicates how steep the line is.
• b - represents the y-intercept of the line, which is where the line crosses the y-axis.

For the equation \[ y = 2 - 3x \], it's already in slope-intercept form. Here, m = -3 and b = 2. This means the slope is -3, indicating the line goes downward from left to right. The y-intercept is 2, meaning the line crosses the y-axis at (0, 2).

Finding Intercepts

Intercepts are points where the line crosses the x or y-axis. They are important for understanding the behavior of the graph.



Finding the y-intercept:

To find the y-intercept, set x to 0 and solve for y:

\[ y = 2 - 3(0) \]

\[ y = 2 \]

Thus, the y-intercept is (0, 2).



Finding the x-intercept:

To find the x-intercept, set y to 0 and solve for x:

\[ 0 = 2 - 3x \]

\[ 3x = 2 \]

\[ x = \frac{2}{3} \]

Thus, the x-intercept is \( \bigg( \frac{2}{3}, 0 \bigg) \).

Graphing Calculator Instructions

Using a graphing calculator can make graphing linear equations easier. Follow these steps to graph the equation:



Step 1: Enter the Equation:

Turn on your calculator and go to the 'Y=' menu. Enter the equation \[ y = 2 - 3x \].



Step 2: Adjust the Viewing Window:

Set the window to show both intercepts. For instance, you could set:

• X_min = -10
• X_max = 10
• Y_min = -10
• Y_max = 10

Step 3: Graph the Equation:

Press 'Graph' to display the graph of \[ y = 2 - 3x \].



Step 4: Find the Intercepts:

Use the 'Calc' function to find the x-intercept (where \( y = 0 \)) and the y-intercept (where \( x = 0 \)).



By following these steps, you can visualize the equation and confirm the intercepts.