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Let's go through this step by step.

### Given quadratic function: 
\[
f(x) = -x^2 + 10x
\]

#### (a) Express \( f(x) \) in standard form.

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

We need to complete the square to convert the given equation into this form.

1. Start with the equation: 
\[
f(x) = -x^2 + 10x
\]

2. Factor out the negative sign from the quadratic and linear terms:
\[
f(x) = -(x^2 - 10x)
\]

3. To complete the square, take half the coefficient of \(x\) (which is -10), square it, and add and subtract this value inside the parentheses:
\[
f(x) = -(x^2 - 10x + 25 - 25)
\]
This simplifies to:
\[
f(x) = -( (x - 5)^2 - 25 )
\]

4. Distribute the negative sign:
\[
f(x) = -(x - 5)^2 + 25
\]

Thus, the standard form is:
\[
f(x) = -(x - 5)^2 + 25
\]

#### (b) Find the vertex, x-intercepts, and y-intercept of \( f \).

1. **Vertex**: From the standard form, we can see that the vertex is \((h, k) = (5, 25)\).

2. **x-intercepts**: To find the x-intercepts, set \( f(x) = 0 \) and solve for \( x \):
\[
-(x - 5)^2 + 25 = 0
\]
\[
(x - 5)^2 = 25
\]
\[
x - 5 = \pm 5
\]
\[
x = 5 + 5 = 10 \quad \text{or} \quad x = 5 - 5 = 0
\]
Thus, the x-intercepts are \( x = 0 \) and \( x = 10 \).

3. **y-intercept**: To find the y-intercept, set \( x = 0 \) in the original equation:
\[
f(0) = -(0)^2 + 10(0) = 0
\]
Thus, the y-intercept is \( y = 0 \).

### Summary:
- Standard form: \( f(x) = -(x - 5)^2 + 25 \)
- Vertex: \( (5, 25) \)
- x-intercepts: \( x = 0 \) and \( x = 10 \)
- y-intercept: \( y = 0 \)

Would you like more details on any step or further explanation?

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Here are 5 related questions:
1. How do you derive the vertex formula from a general quadratic equation?
2. What is the importance of completing the square when working with quadratics?
3. How can we determine the direction of a parabola using the quadratic's coefficient?
4. What are the steps to solve quadratic equations using the quadratic formula?
5. How can the discriminant of a quadratic equation help in determining the number of x-intercepts?

**Tip**: When completing the square, always remember to balance the equation by adding and subtracting the same value inside the parentheses!

A quadratic function f is given: f(x) = -x^2 + 10x. (a) Express f in standard form. (b) Find the vertex and x- and y-intercepts of f.

Learn how to express the quadratic function f(x) = -x^2 + 10x in standard form and find its vertex and intercepts. This problem covers key algebraic concepts such as completing the square and finding the vertex for quadratic functions.

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