ad each question carefuly. Show all work. A ball is thrown from a height of 12 feet above the ground. Atter 1 second, the ball reaches is maximum height of 23 feet above the ground. The path of the ball can be represented by a parabola. a. ( 2 pts) Write a function $$ f x $$ ), in vertex form, for the parabola. b. ( 2 pts) Write the function $$ f(x) $$ in standard form. c. ( 4 pts) What is the height of the ball atter 3 seconds? Explain.

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UpStudy AI · Accepted Answer

To solve the problem, we will follow the steps outlined for each part.

### Given Information:
- The ball is thrown from a height of 12 feet (initial height).
- After 1 second, the ball reaches its maximum height of 23 feet.
- The path of the ball is represented by a parabola.

### Part a: Write a function $$ f(x) $$ in vertex form

The vertex form of a parabola is given by:

$$
f(x) = a(x - h)^2 + k
$$

where $$(h, k)$$ is the vertex of the parabola. From the information provided:
- The vertex is at $$(1, 23)$$ (1 second, maximum height).
- The initial height (when $$x = 0$$) is 12 feet.

To find $$a$$, we can use the point $$(0, 12)$$:

$$
12 = a(0 - 1)^2 + 23
$$

Now, we can solve for $$a$$:

$$
12 = a(1) + 23
$$
$$
12 = a + 23
$$
$$
a = 12 - 23
$$
$$
a = -11
$$

Now we can write the function in vertex form:

$$
f(x) = -11(x - 1)^2 + 23
$$

### Part b: Write the function $$ f(x) $$ in standard form

To convert the vertex form to standard form, we will expand the equation:

$$
f(x) = -11(x - 1)^2 + 23
$$

First, expand $$(x - 1)^2$$:

$$
(x - 1)^2 = x^2 - 2x + 1
$$

Now substitute back into the equation:

$$
f(x) = -11(x^2 - 2x + 1) + 23
$$
$$
= -11x^2 + 22x - 11 + 23
$$
$$
= -11x^2 + 22x + 12
$$

So, the function in standard form is:

$$
f(x) = -11x^2 + 22x + 12
$$

### Part c: What is the height of the ball after 3 seconds?

To find the height of the ball after 3 seconds, we will evaluate $$f(3)$$:

$$
f(3) = -11(3)^2 + 22(3) + 12
$$

Calculating step by step:

1. Calculate $$3^2$$:
   $$
   3^2 = 9
   $$

2. Substitute into the equation:
   $$
   f(3) = -11(9) + 22(3) + 12
   $$

3. Calculate $$-11(9)$$:
   $$
   -11(9) = -99
   $$

4. Calculate $$22(3)$$:
   $$
   22(3) = 66
   $$

5. Now substitute these values back:
   $$
   f(3) = -99 + 66 + 12
   $$

6. Combine the values:
   $$
   f(3) = -99 + 66 = -33
   $$
   $$
   f(3) = -33 + 12 = -21
   $$

Thus, the height of the ball after 3 seconds is:

$$
f(3) = -21 \text{ feet}
$$

### Summary of Answers:
a. $$ f(x) = -11(x - 1)^2 + 23 $$ (vertex form)  
b. $$ f(x) = -11x^2 + 22x + 12 $$ (standard form)  
c. The height of the ball after 3 seconds is -21 feet, which indicates that the ball has already hit the ground before reaching this time.
a. $$ f(x) = -11(x - 1)^2 + 23 $$ b. $$ f(x) = -11x^2 + 22x + 12 $$ c. The ball is -21 feet below the ground after 3 seconds.

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