To analyze the quadratic function \(2x^2 - 9x + 4\), we need to determine its vertex, its zeros, and any potential maximum or minimum values.
- Finding the vertex: The vertex of a quadratic function \(ax^2 + bx + c\) can be found using the formula \(x = -\frac{b}{2a}\).
- Here, \(a = 2\), \(b = -9\).
- So, \(x = -\frac{-9}{2 \cdot 2} = \frac{9}{4} = 2.25\).
Next, we substitute \(x = 2.25\) back into the function to find the corresponding \(y\)-value:
\[ y = 2(2.25)^2 - 9(2.25) + 4 \] \[ = 2(5.0625) - 20.25 + 4 \] \[ = 10.125 - 20.25 + 4 \] \[ = 10.125 - 20.25 + 4 = -6.125 \]
Thus, the function has a minimum at \((-6.125)\) since the coefficient of \(x^2\) is positive.
- Determining the zeros: To find the zeros (x-intercepts) of the function, we can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
- Here, \(b^2 - 4ac = (-9)^2 - 4 \cdot 2 \cdot 4 = 81 - 32 = 49\).
- Thus, the zeros are: \[ x = \frac{9 \pm 7}{4} \]
- So, we have: \[ x = \frac{16}{4} = 4 \quad \text{and} \quad x = \frac{2}{4} = 0.5 \]
Therefore, the zeros of the function are \(0.5\) and \(4\).
Putting it all together, the responses can be evaluated:
- This has a minimum at -6.125. True
- This has a maximum at 4. False
- 0.5 is a zero. True
- 2 is a zero. False
- 4 is a zero. True
So, the correct true statements are:
- This has a minimum at -6.125.
- 0.5 is a zero.
- 4 is a zero.