Question

the equation of the flight path of an object is given by the equation h=4+2x-x^2.
A) calculate the maximum height of an object and illustrate with a graph the path travel by an object?

Answer 1

To determine the maximum height of the flight path described by the equation \( h = 4 + 2x - x^2 \), we can recognize that this equation is a quadratic function in the standard form \( h = ax^2 + bx + c \). Here, \( a = -1 \), \( b = 2 \), and \( c = 4 \).

### A) Finding the Maximum Height

1. **Finding the Vertex**:
The maximum height of a quadratic function occurs at its vertex. The x-coordinate of the vertex can be calculated using the formula:

\[
x = -\frac{b}{2a}
\]

Substituting the values of \( a \) and \( b \):

\[
x = -\frac{2}{2(-1)} = 1
\]

2. **Calculating the Maximum Height**:
We substitute \( x = 1 \) back into the equation to find the height \( h \):

\[
h = 4 + 2(1) - (1)^2
\]
\[
h = 4 + 2 - 1 = 5
\]

Thus, the maximum height of the object is **5 units**.

### Graphing the Flight Path

To illustrate the flight path, we can plot the quadratic function \( h = 4 + 2x - x^2 \).

1. **Finding additional points**:
To have a clearer graph, we can compute a few more points by selecting values for \( x \):

- If \( x = 0 \):
\[
h = 4 + 2(0) - (0)^2 = 4
\]

- If \( x = 2 \):
\[
h = 4 + 2(2) - (2)^2 = 4 + 4 - 4 = 4
\]

- If \( x = 3 \):
\[
h = 4 + 2(3) - (3)^2 = 4 + 6 - 9 = 1
\]

- If \( x = -1 \):
\[
h = 4 + 2(-1) - (-1)^2 = 4 - 2 - 1 = 1
\]

We can summarize the calculated points:
- \( (0, 4) \)
- \( (1, 5) \) - maximum height
- \( (2, 4) \)
- \( (3, 1) \)
- \( (-1, 1) \)

2. **Plotting the graph**:
Below is a simple representation of what the graph would look like:

```
h
| *
| * *
| * *
| * *
| * *
| * *
|* *
----------------------- x
-1 0 1 2 3
```

The curve opens downwards (since \( a < 0 \)), with the vertex at the point \( (1, 5) \), which represents the maximum height.