15.	In the figure, the vertex of the graph of a quadratic function y = f(x) is	y
(3 , –10) and the y-intercept is 8. Find f(x).


To find the quadratic function \( f(x) \) given its vertex and y-intercept, we will use the vertex form of a quadratic function and the given 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: \((3, -10)\)
- Y-intercept: \((0, 8)\)

First, plug in the vertex into the vertex form:
\[ f(x) = a(x - 3)^2 - 10 \]

Next, use the y-intercept to find the value of \(a\). The y-intercept means when \(x = 0\), \( f(x) = 8 \). Plug these values into the function:
\[ 8 = a(0 - 3)^2 - 10 \]
\[ 8 = a(9) - 10 \]
\[ 8 = 9a - 10 \]
\[ 8 + 10 = 9a \]
\[ 18 = 9a \]
\[ a = 2 \]

So the quadratic function is:
\[ f(x) = 2(x - 3)^2 - 10 \]

Let's expand this to confirm it fits the given conditions:
\[ f(x) = 2(x^2 - 6x + 9) - 10 \]
\[ f(x) = 2x^2 - 12x + 18 - 10 \]
\[ f(x) = 2x^2 - 12x + 8 \]

Thus, the quadratic function is:
\[ f(x) = 2x^2 - 12x + 8 \]

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

1. How do you find the roots of a quadratic function given its standard form?
2. What are the steps to complete the square for a quadratic function?
3. How do you determine if a quadratic function opens upwards or downwards?
4. What is the vertex form of a quadratic function and why is it useful?
5. How can you use the quadratic formula to solve real-world problems?

**Tip:** When converting a quadratic function from vertex form to standard form, carefully expand the squared term and combine like terms to ensure accuracy.

Learn how to find the quadratic function y = f(x) given its vertex and y-intercept. This problem involves using the vertex form of a quadratic function and solving for the coefficients using specific data points. Suitable for students in Grades 9-10.

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