Sure!
### Part a: Graph the Equation
The given linear equation is
[tex]$ y = -10x + 3000 $[/tex]
where:
- [tex]\( y \)[/tex] represents the height of the skydiver in feet.
- [tex]\( x \)[/tex] represents the time in seconds.
To graph the equation, follow these steps:
1. Determine the intercepts:
- Y-intercept: This is the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex].
Substituting [tex]\( x = 0 \)[/tex] into the equation:
[tex]$
y = -10(0) + 3000 = 3000
$[/tex]
So the y-intercept is (0, 3000).
- X-intercept: This is the value of [tex]\( x \)[/tex] when [tex]\( y = 0 \)[/tex].
Substituting [tex]\( y = 0 \)[/tex] into the equation:
[tex]$
0 = -10x + 3000
$[/tex]
Solving for [tex]\( x \)[/tex]:
[tex]$
10x = 3000
$[/tex]
[tex]$
x = 300
$[/tex]
So the x-intercept is (300, 0).
2. Plot these points on a coordinate plane:
- The y-intercept is at (0, 3000).
- The x-intercept is at (300, 0).
3. Draw the line passing through these intercepts.
### Interpretation of the [tex]$x$[/tex]-Intercept and Slope
- X-Intercept: The x-intercept is [tex]\( (300, 0) \)[/tex]. This point means that after 300 seconds (or 5 minutes), the height of the skydiver will be 0 feet, which means the skydiver has reached the ground.
- Slope: The slope of the line is the coefficient of [tex]\( x \)[/tex] in the equation [tex]\( y = -10x + 3000 \)[/tex], which is -10. The slope represents the rate of change of height with respect to time. A slope of -10 means that the skydiver descends at a rate of 10 feet per second. This negative slope indicates that the height is decreasing over time.
### Summary
- Graphing the Equation: You plot the points (0, 3000) and (300, 0), then draw the line through these points.
- X-Intercept (300, 0): After 300 seconds, the skydiver reaches the ground.
- Slope (-10): The skydiver’s height decreases by 10 feet every second.
