Problem 62 - 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. $$y=800 x+8$$

Step-by-Step Solution

Answer

The x-intercept is \( x = -0.01 \) and the y-intercept is \( y = 8 \).

Step 1 - Enter the Equation

Enter the given equation, \( y = 800x + 8 \), into the graphing calculator. Usually, there's a button for 'y=' or an equivalent function to input the equation.

Step 2 - Set the Window

Adjust the window settings to ensure both intercepts are visible. A good starting point for the x-range could be from -0.01 to 0.01 and for the y-range from -10 to 10.

Step 3 - Graph the Equation

After setting the window, graph the equation to visualize it.

Step 4 - Locate the X-Intercept

Use the 'zero' or 'root' feature of the calculator to find the x-intercept, where the graph crosses the x-axis. This occurs when \( y = 0 \).

Step 5 - Calculate the X-Intercept

Set \( y = 0 \) in the equation: \( 0 = 800x + 8 \). Solve for \( x \), which gives \( x = -\frac{8}{800} = -0.01 \).

Step 6 - Locate the Y-Intercept

Use the 'value' feature of the calculator to find the y-intercept, where the graph crosses the y-axis. This occurs when \( x = 0 \).

Step 7 - Calculate the Y-Intercept

Set \( x = 0 \) in the equation: \( y = 800(0) + 8 \). This gives \( y = 8 \).

Step 8 - Verify on Graph

Confirm the intercepts by visually checking the points where the graph crosses the axes.

Key Concepts

x-intercept

The x-intercept is the point where a graph crosses the x-axis. This means that at this point, the value of y is always zero. To find the x-intercept, you can follow these steps:

• Set y to 0 in the equation of the line.
• Solve the equation for x.

For instance, in the equation provided,
\( y = 800x + 8 \)
You set y to 0:
\[ 0 = 800x + 8 \]
Next, solve for x:
\[ x = -\frac{8}{800} = -0.01 \]
This means the x-intercept is at (-0.01, 0). If using a graphing calculator, you can also use the 'zero' or 'root' feature to find the x-intercept visually.

y-intercept

The y-intercept is the point where the graph crosses the y-axis. At this point, the value of x is always zero. Here's how you can find the y-intercept:

• Set x to 0 in the equation of the line.
• Solve the equation for y.

Using the same equation:
\( y = 800x + 8 \)
Set x to 0:
\[ y = 800(0) + 8 \]
Thus, y equals 8. So, the y-intercept is at (0, 8). Your graphing calculator can assist with this too. Use the 'value' feature to input x = 0 and see the y-value directly.

equation of a line

An equation of a line in slope-intercept form is written as \( y = mx + b \). Here:

• \( m \) represents the slope, which shows how steep the line is.
• \( b \) is the y-intercept, indicating where the line crosses the y-axis.

In our example:
\( y = 800x + 8 \)
The slope, \( m \), is 800, and the y-intercept, \( b \), is 8.
The slope tells us that for every 1 unit increase in x, y increases by 800 units. This makes the line very steep. Plotting such lines using a graphing calculator is helpful due to its precision. Remember:

• Use the 'y=' function to input the equation.
• Adjust the window settings to visualize the intercepts.

This helps in understanding the behavior and position of the line on the graph.