Question

A rock is thrown from a cliff up into the air from a height of 4.95 meters. After 1.5 seconds, the rock reaches a maximum height of 16.2 meters. It then begins to fall and hits the ground 3.3 seconds after it is thrown.
Let f(x) be the height (in meters) of the rock x seconds after it is thrown. Then, the function fis quadratic.
(Its graph is a parabola.) Write an equation for the quadratic function f.

Answer 1

To write a quadratic function \( f(x) \) that models the height of the rock over time, we start with the general form of a quadratic equation:

\[
f(x) = ax^2 + bx + c
\]

Where:
- \( a \), \( b \), and \( c \) are constants we need to determine.
- \( x \) is the time in seconds after the rock is thrown.
- \( f(x) \) is the height in meters.

Given the problem, we have the following information:

1. The rock starts from a height of \( 4.95 \) meters at \( x = 0 \):
\[
f(0) = c = 4.95
\]

2. The maximum height occurs at \( x = 1.5 \) seconds, where \( f(1.5) = 16.2 \).

3. The rock hits the ground at \( x = 3.3 \) seconds, so \( f(3.3) = 0 \).

We can create a system of equations based on this information.

### Step 1: Create the equations

Using the maximum height and the height at the initial and final time, we can write:

1. From the initial height:
\[
c = 4.95
\]

2. From the maximum height:
\[
f(1.5) = a(1.5)^2 + b(1.5) + 4.95 = 16.2
\]
Simplifying this:
\[
2.25a + 1.5b + 4.95 = 16.2 \implies 2.25a + 1.5b = 11.25 \tag{1}
\]

3. From the final height:
\[
f(3.3) = a(3.3)^2 + b(3.3) + 4.95 = 0
\]
Simplifying this:
\[
10.89a + 3.3b + 4.95 = 0 \implies 10.89a + 3.3b = -4.95 \tag{2}
\]

### Step 2: Solve the system of equations

From equation (1):
\[
2.25a + 1.5b = 11.25
\]
And equation (2):
\[
10.89a + 3.3b = -4.95
\]

We can express \( b \) in terms of \( a \) from equation (1):
\[
1.5b = 11.25 - 2.25a \implies b = \frac{11.25 - 2.25a}{1.5} = 7.5 - 1.5a \tag{3}
\]

Substituting equation (3) into equation (2):
\[
10.89a + 3.3(7.5 - 1.5a) = -4.95
\]
Expanding this:
\[
10.89a + 24.75 - 4.95a = -4.95
\]
Combining like terms:
\[
5.94a + 24.75 = -4.95
\]
Subtracting \( 24.75 \) from both sides:
\[
5.94a = -29.7 \implies a = \frac{-29.7}{5.94} \approx -5
\]

Now substitute \( a \) back into equation (3) to find \( b \):
\[
b = 7.5 - 1.5(-5) = 7.5 + 7.5 = 15
\]

### Step 3: Form the quadratic function

Now we have:
- \( a \approx -5 \)
- \( b = 15 \)
- \( c = 4.95 \)

Thus, the quadratic function is:
\[
f(x) = -5x^2 + 15x + 4.95
\]

### Final Answer
The equation for the height of the rock over time is:

\[
f(x) = -5x^2 + 15x + 4.95
\]