Question

Select all options below that are true for the characteristics of 2x2−9x+4
.(3 points)
Responses

0.5 is a zero.
0.5 is a zero.

2 is a zero.
2 is a zero.

This has a minimum at -6.125.
This has a minimum at -6.125.

This has a maximum at 4.
This has a maximum at 4.

4 is a zero.

Answer 1

To evaluate the characteristics of the quadratic function \( f(x) = 2x^2 - 9x + 4 \), we need to analyze it step-by-step.

1. **Finding the zeros (roots) of the quadratic function:**
We can find the zeros using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = -9 \), and \( c = 4 \).

First, we calculate the discriminant:
\[
b^2 - 4ac = (-9)^2 - 4 \cdot 2 \cdot 4 = 81 - 32 = 49
\]

Since the discriminant is positive, there are two real roots. Now we apply the quadratic formula:
\[
x = \frac{-(-9) \pm \sqrt{49}}{2 \cdot 2} = \frac{9 \pm 7}{4}
\]

This gives us two solutions:
\[
x_1 = \frac{16}{4} = 4 \quad \text{and} \quad x_2 = \frac{2}{4} = 0.5
\]

So, **0.5 is a zero** and **4 is a zero**.

2. **Finding the vertex (minimum or maximum):**
The vertex of a quadratic function \( ax^2 + bx + c \) occurs at \( x = -\frac{b}{2a} \):
\[
x = -\frac{-9}{2 \cdot 2} = \frac{9}{4} = 2.25
\]

To find the minimum value, we evaluate \( f(2.25) \):
\[
f(2.25) = 2(2.25)^2 - 9(2.25) + 4
\]
First, calculate \( (2.25)^2 = 5.0625 \):
\[
f(2.25) = 2(5.0625) - 20.25 + 4 = 10.125 - 20.25 + 4 = -6.125
\]
This confirms that the vertex represents a **minimum** at \(-6.125\).

3. **Determining if it has a maximum:**
Since the coefficient of \( x^2 \) (which is 2) is positive, this quadratic opens upwards. Therefore, there is no maximum.

### Summary
- 0.5 is a zero. **(True)**
- 2 is a zero. **(False)**
- This has a minimum at -6.125. **(True)**
- This has a maximum at 4. **(False)**
- 4 is a zero. **(True)**

### The true options are:
- **0.5 is a zero.**
- **This has a minimum at -6.125.**
- **4 is a zero.**