18.	The y-intercept of the graph of a quadratic function y = f(x) is 8. If the graph touches the x-axis at x = 1 only, find f(x).

Given that the y-intercept of the quadratic function \( y = f(x) \) is 8, and the graph touches the x-axis at \( x = 1 \) only (indicating that it has a double root at \( x = 1 \)), we can express the quadratic function in the following form:

\[ f(x) = a(x - 1)^2 \]

Since the y-intercept is 8, when \( x = 0 \), \( y = 8 \):

\[ f(0) = 8 \]

Substitute \( x = 0 \) into the equation:

\[ 8 = a(0 - 1)^2 \]

Simplify inside the parentheses:

\[ 8 = a(1)^2 \]

\[ 8 = a \]

So, \( a = 8 \).

Thus, the quadratic function is:

\[ f(x) = 8(x - 1)^2 \]

This is the function that satisfies the given conditions.

Would you like more details or have any questions?

Here are some additional questions you might find useful:

1. How can you determine the vertex of a quadratic function from its standard form?
2. What does it mean for a quadratic function to have a double root?
3. How do you find the x-intercepts of a quadratic function?
4. How does the y-intercept affect the graph of a quadratic function?
5. What are the properties of the parabola represented by a quadratic function?

**Tip:** When a quadratic function has a double root, it indicates that the vertex of the parabola lies on the x-axis, making it either a minimum or maximum point depending on the direction of the parabola's opening.

Learn how to find a quadratic function f(x) given its y-intercept and a double root at x = 1. This problem involves quadratic functions, vertex form, and understanding the impact of double roots on the graph. Suitable for students in Grades 9-12.

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