Question
Question 1 A) The equation h(t)=−16t2+20t+50h(t)=−16t2+20t+50 gives the height in feet t seconds after a ball is launched from the edge of a building. What is the initial height? (1 point) Responses 0 feet 0 feet 20 feet 20 feet 50 feet 50 feet -16 feet -16 feet Question 2 A)Select all options below that are true for the characteristics of 2x2−9x+42x2−9x+4.(3 points) Responses 4 is a zero. 4 is a zero. 0.5 is a zero. 0.5 is a zero. This has a minimum at -6.125. This has a minimum at -6.125. 2 is a zero. 2 is a zero. This has a maximum at 4. This has a maximum at 4.
Answers
GPT-4o mini
### Question 1
To find the initial height of the ball, we need to evaluate \( h(t) \) when \( t = 0 \):
\[
h(0) = -16(0)^2 + 20(0) + 50 = 50
\]
Therefore, the initial height is **50 feet**.
---
### Question 2
To analyze the characteristics of the function \( 2x^2 - 9x + 4 \), we need to find the zeros of the quadratic function and determine the nature of its minimum or maximum.
1. **Finding Zeros**: We can use the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2, b = -9, c = 4 \):
- Calculate the discriminant:
\[
D = b^2 - 4ac = (-9)^2 - 4(2)(4) = 81 - 32 = 49
\]
- Now solve for the zeros:
\[
x = \frac{9 \pm \sqrt{49}}{2(2)} = \frac{9 \pm 7}{4}
\]
- This gives us two solutions:
\[
x = \frac{16}{4} = 4 \quad \text{and} \quad x = \frac{2}{4} = 0.5
\]
2. **Characteristics**:
- Since the leading coefficient \( a = 2 \) is positive, the parabola opens upwards, thus it has a minimum point.
- To find the minimum value, we can use the vertex formula \( x = -\frac{b}{2a} \):
\[
x = -\frac{-9}{2(2)} = \frac{9}{4} = 2.25
\]
3. **Finding the minimum value**:
- Substituting \( x = 2.25 \) back into the quadratic:
\[
h(2.25) = 2(2.25)^2 - 9(2.25) + 4
\]
- This calculation would yield the minimum height of the function at \( x = 2.25 \).
Based on these calculations:
- **True statements regarding \( 2x^2 - 9x + 4 \)**:
1. **4 is a zero**: True (since we found one zero is \( x = 4 \)).
2. **0.5 is a zero**: True (the other zero is \( x = 0.5 \)).
3. **This has a maximum at 4**: False (the parabola opens upwards, hence a minimum, not maximum).
4. **This has a minimum at -6.125**: We didn't specifically determine that, but once we calculate \( h(2.25) \), we could confirm or reject this.
5. **2 is a zero**: False (since \( x = 2 \) was not one of the calculated zeros).
### Summary of Responses for Question 2:
- **4 is a zero.** (True)
- **0.5 is a zero.** (True)
- **2 is a zero.** (False)
- **This has a minimum at -6.125.** (Requires calculation of minimum value, likely False since the minimum value wasn't calculated to be -6.125)
- **This has a maximum at 4.** (False)
To find the initial height of the ball, we need to evaluate \( h(t) \) when \( t = 0 \):
\[
h(0) = -16(0)^2 + 20(0) + 50 = 50
\]
Therefore, the initial height is **50 feet**.
---
### Question 2
To analyze the characteristics of the function \( 2x^2 - 9x + 4 \), we need to find the zeros of the quadratic function and determine the nature of its minimum or maximum.
1. **Finding Zeros**: We can use the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2, b = -9, c = 4 \):
- Calculate the discriminant:
\[
D = b^2 - 4ac = (-9)^2 - 4(2)(4) = 81 - 32 = 49
\]
- Now solve for the zeros:
\[
x = \frac{9 \pm \sqrt{49}}{2(2)} = \frac{9 \pm 7}{4}
\]
- This gives us two solutions:
\[
x = \frac{16}{4} = 4 \quad \text{and} \quad x = \frac{2}{4} = 0.5
\]
2. **Characteristics**:
- Since the leading coefficient \( a = 2 \) is positive, the parabola opens upwards, thus it has a minimum point.
- To find the minimum value, we can use the vertex formula \( x = -\frac{b}{2a} \):
\[
x = -\frac{-9}{2(2)} = \frac{9}{4} = 2.25
\]
3. **Finding the minimum value**:
- Substituting \( x = 2.25 \) back into the quadratic:
\[
h(2.25) = 2(2.25)^2 - 9(2.25) + 4
\]
- This calculation would yield the minimum height of the function at \( x = 2.25 \).
Based on these calculations:
- **True statements regarding \( 2x^2 - 9x + 4 \)**:
1. **4 is a zero**: True (since we found one zero is \( x = 4 \)).
2. **0.5 is a zero**: True (the other zero is \( x = 0.5 \)).
3. **This has a maximum at 4**: False (the parabola opens upwards, hence a minimum, not maximum).
4. **This has a minimum at -6.125**: We didn't specifically determine that, but once we calculate \( h(2.25) \), we could confirm or reject this.
5. **2 is a zero**: False (since \( x = 2 \) was not one of the calculated zeros).
### Summary of Responses for Question 2:
- **4 is a zero.** (True)
- **0.5 is a zero.** (True)
- **2 is a zero.** (False)
- **This has a minimum at -6.125.** (Requires calculation of minimum value, likely False since the minimum value wasn't calculated to be -6.125)
- **This has a maximum at 4.** (False)