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?

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!