Question
A rocket's path is modeled by the equation y=−2x^2 +4x+6 . Graph the function. Identify the x-intercepts and interpret their meaning in the context of the rocket launch.(2 points) x-intercepts (list smaller one first): and Question 2 Using the context, interpret the meaning of the above x-intercept(s).(2 points)
Answers
GPT-4o mini
To analyze the function \( y = -2x^2 + 4x + 6 \), let's first find the x-intercepts by setting \( y = 0 \):
\[
0 = -2x^2 + 4x + 6
\]
We can rearrange the equation:
\[
-2x^2 + 4x + 6 = 0
\]
Divide the entire equation by -2 to simplify:
\[
x^2 - 2x - 3 = 0
\]
Next, we factor the quadratic:
\[
(x - 3)(x + 1) = 0
\]
Setting each factor to zero gives the x-intercepts:
\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]
\[
x + 1 = 0 \quad \Rightarrow \quad x = -1
\]
Thus, the x-intercepts are \( x = -1 \) and \( x = 3 \).
### Question 1: X-intercepts
**X-Intercepts:**
- Smaller one first: \( -1, 3 \)
### Question 2: Interpretation of the X-Intercepts
The x-intercepts represent the points in time (in whatever time unit is chosen) when the rocket has a height of \( y = 0 \), meaning it is at ground level.
1. **X-Intercept \( x = -1 \)**: This intercept occurs before the launch at \( x = -1 \). In the context of the rocket launch, this could represent a theoretical scenario or extension of the model where the rocket has not yet been launched (for instance, if time is represented in seconds, this could correlate to a situation that doesn't correspond to a real launch).
2. **X-Intercept \( x = 3 \)**: This intercept represents the rocket returning to the ground at \( x = 3 \) seconds after launch. This means the rocket reaches its maximum height and then descends back to ground level approximately 3 seconds after being launched.
Overall, the x-intercepts inform us not only about the specific moments when the rocket is at ground level but also provide insight into the duration of the rocket's flight.
\[
0 = -2x^2 + 4x + 6
\]
We can rearrange the equation:
\[
-2x^2 + 4x + 6 = 0
\]
Divide the entire equation by -2 to simplify:
\[
x^2 - 2x - 3 = 0
\]
Next, we factor the quadratic:
\[
(x - 3)(x + 1) = 0
\]
Setting each factor to zero gives the x-intercepts:
\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]
\[
x + 1 = 0 \quad \Rightarrow \quad x = -1
\]
Thus, the x-intercepts are \( x = -1 \) and \( x = 3 \).
### Question 1: X-intercepts
**X-Intercepts:**
- Smaller one first: \( -1, 3 \)
### Question 2: Interpretation of the X-Intercepts
The x-intercepts represent the points in time (in whatever time unit is chosen) when the rocket has a height of \( y = 0 \), meaning it is at ground level.
1. **X-Intercept \( x = -1 \)**: This intercept occurs before the launch at \( x = -1 \). In the context of the rocket launch, this could represent a theoretical scenario or extension of the model where the rocket has not yet been launched (for instance, if time is represented in seconds, this could correlate to a situation that doesn't correspond to a real launch).
2. **X-Intercept \( x = 3 \)**: This intercept represents the rocket returning to the ground at \( x = 3 \) seconds after launch. This means the rocket reaches its maximum height and then descends back to ground level approximately 3 seconds after being launched.
Overall, the x-intercepts inform us not only about the specific moments when the rocket is at ground level but also provide insight into the duration of the rocket's flight.