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.

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.