https://file.aiquickdraw.com/mathbot/file/1727497540357xpclrtf2.jpg

The equation given in the image is \( f(x) = -4x^2 + 10x + 4 \). Let's find the required parameters for the parabola defined by this equation.

### A) The Vertex

The vertex form of a quadratic equation \( ax^2 + bx + c \) is given by \( (h, k) \) where:

\[ h = -\frac{b}{2a} \]
\[ k = f(h) \]

Substituting \( a = -4 \) and \( b = 10 \):

\[ h = -\frac{10}{2 \times -4} = 1.25 \]
\[ k = -4(1.25)^2 + 10(1.25) + 4 \]

We need to compute \( k \):

### B) The Vertical Intercept

The vertical intercept is the point where \( x = 0 \), which is \( f(0) \). Directly substituting in the equation gives:

\[ f(0) = -4(0)^2 + 10(0) + 4 = 4 \]

### C) The x-Intercepts

The x-intercepts are the values of \( x \) where \( f(x) = 0 \):

\[ -4x^2 + 10x + 4 = 0 \]

We can use the quadratic formula to find these:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Substituting \( a = -4 \), \( b = 10 \), and \( c = 4 \):

Let's compute these values using the equations provided.Here are the results for the parabola described by the equation \( f(x) = -4x^2 + 10x + 4 \):

### A) The Vertex
The vertex of the parabola is at \( (1.25, 10.25) \).

### B) The Vertical Intercept
The vertical intercept of the parabola is the point \( (0, 4) \).

### C) The x-Intercepts
The x-intercepts of the parabola are approximately at \( x = 1.25 - \frac{\sqrt{41}}{4} \) and \( x = 1.25 + \frac{\sqrt{41}}{4} \). These values can be approximated as:
- \( x \approx -0.276 \)
- \( x \approx 2.776 \)

Would you like detailed steps on how to derive any of these results, or do you have any other questions on this topic?

Here are five related questions you might consider:
1. How does changing the coefficient \( a \) in the quadratic equation affect the shape of the parabola?
2. What is the axis of symmetry for a quadratic function, and how is it calculated?
3. How can the discriminant of a quadratic equation be used to determine the number of x-intercepts?
4. What happens to the graph of the quadratic function if the sign of \( a \) is reversed?
5. How would adding a constant term to the quadratic function \( f(x) = -4x^2 + 10x + 4 \) affect its graph?

**Tip**: Remember that the vertex form of a quadratic equation \( y = a(x-h)^2 + k \) can be very useful for graphing and understanding the properties of the parabola, such as its maximum or minimum value and its symmetry.

Consider the parabola given by the equation f(x) = -4x^2 + 10x + 4. Find the following for this parabola: A) The vertex, B) The vertical intercept is the point, C) Find the coordinates of the two x-intercepts of the parabola and write them as a list, separated by commas.

Learn how to find the vertex, vertical intercept, and x-intercepts of a parabola given by the quadratic equation f(x) = -4x^2 + 10x + 4. This detailed step-by-step solution is perfect for students in Grades 9-11 and covers key algebraic concepts such as the quadratic formula and the properties of parabolas.

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