14.	In the figure, the vertex of the graph of a quadratic function y = f(x) is (–2 , 5) and the y-intercept is –3. Find f(x).

To find the quadratic function \( f(x) \) given its vertex and y-intercept, we'll use the vertex form of a quadratic function and some additional information.

The vertex form of a quadratic function is:
\[ f(x) = a(x - h)^2 + k \]
where \((h, k)\) is the vertex of the parabola.

Given:
- Vertex: \((-2, 5)\)
- Y-intercept: \((0, -3)\)

First, plug in the vertex into the vertex form:
\[ f(x) = a(x + 2)^2 + 5 \]

Next, use the y-intercept to find the value of \(a\). The y-intercept means when \(x = 0\), \( f(x) = -3 \). Plug these values into the function:
\[ -3 = a(0 + 2)^2 + 5 \]
\[ -3 = a(4) + 5 \]
\[ -3 = 4a + 5 \]
\[ -3 - 5 = 4a \]
\[ -8 = 4a \]
\[ a = -2 \]

So the quadratic function is:
\[ f(x) = -2(x + 2)^2 + 5 \]

Let's expand this to confirm it fits the given conditions:
\[ f(x) = -2(x^2 + 4x + 4) + 5 \]
\[ f(x) = -2x^2 - 8x - 8 + 5 \]
\[ f(x) = -2x^2 - 8x - 3 \]

Thus, the quadratic function is:
\[ f(x) = -2x^2 - 8x - 3 \]

Would you like more details or have any questions? Here are some additional questions you might find interesting:

1. How do you find the vertex of a quadratic function given its standard form?
2. What are the steps to convert a quadratic function from standard form to vertex form?
3. How do you determine the axis of symmetry of a quadratic function?
4. What is the significance of the discriminant in a quadratic equation?
5. Can you derive the quadratic formula from the general form of a quadratic equation?

**Tip:** To quickly determine the vertex form from the standard form \(ax^2 + bx + c\), use the formula for the vertex: \(h = -\frac{b}{2a}\) and \(k = f(h)\).

Learn how to find the quadratic function f(x) given its vertex at (-2, 5) and y-intercept -3. Step-by-step solution using vertex form and y-intercept information. Suitable for high school students studying quadratic functions.

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