Problem 66 - 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. $$y=30.6-3.6 x$$

Step-by-Step Solution

Answer

The x-intercept is at (8.5, 0) and the y-intercept is at (0, 30.6).

Step 1 - Enter the Equation

Enter the equation \(y = 30.6 - 3.6x\) into your graphing calculator. Most calculators have a Y= button that allows you to input equations for graphing.

Step 2 - Set the Graphing Window

Adjust the window settings to ensure both intercepts are visible. For this equation, choose an appropriate range for X and Y. A good starting point could be: \(X_{min} = -10\), \(X_{max} = 10\), \(Y_{min} = -10\), \(Y_{max} = 40\).

Step 3 - Graph the Equation

Graph the equation using the graphing button on your calculator. This will display the line on the screen.

Step 4 - Find the X-Intercept

Use the 'zero' or 'root' feature of your calculator to find the x-intercept. This feature helps you determine where the line crosses the x-axis. Move the cursor near the intercept and select it to find \(x\) when \(y=0\).

Step 5 - Find the Y-Intercept

The y-intercept occurs when \(x = 0\). You can either see this directly from the graph or substitute \(x = 0\) into the equation to find \(y\). On the calculator, simply check the value at \(x=0\).

Key Concepts

Graphing Calculator

A graphing calculator is an essential tool for plotting and analyzing linear equations. To graph an equation like \(y = 30.6 - 3.6x\), you will need to enter it into the calculator using the Y= button. This allows you to input your equation so that the graphing feature can visualize it. Once the equation is input, you can use your calculator’s various functions to explore its properties. These calculators are particularly useful because they allow you to adjust the graphing window to focus on specific regions and identify intercepts accurately. In this exercise, using a graphing calculator will make finding the x-intercept and y-intercept much simpler.

X-Intercept

The x-intercept of a graph is the point where the line crosses the x-axis. For the equation \(y = 30.6 - 3.6x\), this occurs when \(y = 0\). To find this intercept using a graphing calculator, you can use the 'zero' or 'root' feature. Here’s how you can do it:

• Graph the equation.
• Access the 'zero' or 'root' function from the calculator's menu (often found under the 'Calc' menu).
• Move the cursor near where the line crosses the x-axis and select the zero/root option.

This will give you the x-value where \(y = 0\), showing you the exact x-intercept. For our equation, use your calculator to find the exact value where the line intersects the x-axis.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis, which occurs when \(x = 0\). For the equation \(y = 30.6 - 3.6x\), substitute \(x = 0\) directly into the equation:
\ y = 30.6 - 3.6(0) \ y = 30.6
This means the y-intercept is at the point (0, 30.6). On a graphing calculator, you can visually confirm this by looking at the graph where it crosses the y-axis. Additionally, many calculators allow you to directly input \(x=0\) to find the corresponding \(y\)-value.

Window Settings

Window settings on your graphing calculator are crucial for effectively viewing graphs, especially for identifying intercepts. If the defaults aren't appropriate, the intercepts might be off-screen. For the equation \(y = 30.6 - 3.6x\), here's a suggested setting:
\ X_{min} = -10, X_{max} = 10 \ Y_{min} = -10, Y_{max} = 40
These settings ensure that both intercepts are visible on the screen.

• Adjust the X and Y ranges based on the equation's possible values.
• If the graph seems too compressed or elongated, tweak the settings gently.
• Always check to ensure the intercepts are visible without distortion.

Proper window settings help to accurately analyze your graph and find intercepts efficiently.