Question
Select all options below that are true for the characteristics of 2x2−9x+4
.(3 points)
Responses
2 is a zero.
2 is a zero.
0.5 is a zero.
0.5 is a zero.
This has a maximum at 4.
This has a maximum at 4.
This has a minimum at -6.125.
This has a minimum at -6.125.
4 is a zero.
.(3 points)
Responses
2 is a zero.
2 is a zero.
0.5 is a zero.
0.5 is a zero.
This has a maximum at 4.
This has a maximum at 4.
This has a minimum at -6.125.
This has a minimum at -6.125.
4 is a zero.
Answers
GPT-4o mini
To analyze the quadratic function \(2x^2 - 9x + 4\), we can determine its characteristics, including the zeros and whether it has a maximum or minimum.
1. **Finding the zeros**: We can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 2\), \(b = -9\), and \(c = 4\).
- Calculate the discriminant:
\[
b^2 - 4ac = (-9)^2 - 4(2)(4) = 81 - 32 = 49
\]
- Apply the quadratic formula:
\[
x = \frac{-(-9) \pm \sqrt{49}}{2(2)} = \frac{9 \pm 7}{4}
\]
This gives us two zeros:
\[
x = \frac{16}{4} = 4 \quad \text{and} \quad x = \frac{2}{4} = 0.5
\]
So, **0.5 and 4 are zeros** of the function.
2. **Determining maximum or minimum**: The coefficient of \(x^2\) is positive (\(2 > 0\)), so the parabola opens upwards. This means the function has a minimum, not a maximum.
- To find the minimum value, we need to find the vertex using the formula \(x_{\text{vertex}} = \frac{-b}{2a}\):
\[
x_{\text{vertex}} = \frac{-(-9)}{2(2)} = \frac{9}{4} = 2.25
\]
- Next, substitute \(x = 2.25\) back into the expression to find the minimum value:
\[
f(2.25) = 2(2.25)^2 - 9(2.25) + 4 = 2(5.0625) - 20.25 + 4 = 10.125 - 20.25 + 4 = -6.125
\]
From the analysis above, we can summarize the characteristics of the quadratic function \(2x^2 - 9x + 4\):
- **Zeros**: 0.5 and 4 are indeed zeros.
- **Minimum**: This function has a minimum at \(-6.125\) and **not a maximum**.
**True responses:**
- 0.5 is a zero.
- 4 is a zero.
- This has a minimum at -6.125.
**False responses**:
- 2 is a zero. (False)
- This has a maximum at 4. (False)
1. **Finding the zeros**: We can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 2\), \(b = -9\), and \(c = 4\).
- Calculate the discriminant:
\[
b^2 - 4ac = (-9)^2 - 4(2)(4) = 81 - 32 = 49
\]
- Apply the quadratic formula:
\[
x = \frac{-(-9) \pm \sqrt{49}}{2(2)} = \frac{9 \pm 7}{4}
\]
This gives us two zeros:
\[
x = \frac{16}{4} = 4 \quad \text{and} \quad x = \frac{2}{4} = 0.5
\]
So, **0.5 and 4 are zeros** of the function.
2. **Determining maximum or minimum**: The coefficient of \(x^2\) is positive (\(2 > 0\)), so the parabola opens upwards. This means the function has a minimum, not a maximum.
- To find the minimum value, we need to find the vertex using the formula \(x_{\text{vertex}} = \frac{-b}{2a}\):
\[
x_{\text{vertex}} = \frac{-(-9)}{2(2)} = \frac{9}{4} = 2.25
\]
- Next, substitute \(x = 2.25\) back into the expression to find the minimum value:
\[
f(2.25) = 2(2.25)^2 - 9(2.25) + 4 = 2(5.0625) - 20.25 + 4 = 10.125 - 20.25 + 4 = -6.125
\]
From the analysis above, we can summarize the characteristics of the quadratic function \(2x^2 - 9x + 4\):
- **Zeros**: 0.5 and 4 are indeed zeros.
- **Minimum**: This function has a minimum at \(-6.125\) and **not a maximum**.
**True responses:**
- 0.5 is a zero.
- 4 is a zero.
- This has a minimum at -6.125.
**False responses**:
- 2 is a zero. (False)
- This has a maximum at 4. (False)