A rocket is fired into the air with an initial velocity of 98 m/s. The height ( h ) of the rocket after t seconds is given by the expression h = 98t - 4.9t2.

a. What the average rate of change over the first 2 seconds.
b. At what point does the rocket reach its maximum height? Show a graphical and algebraic solution.
c. Over what intervals is the rocket’s height increasing and decreasing?

(a) avg speed is (f(2)-f(0))/2 - distance/time

(b) as with any quadratic, the vertex is at x = -b/2a
(c) v = 98-9.8t
height is increasing where v>0

a. find the heights at t = 0 and t = 2, then perform a "slope" calculation of

average rate of change = change in distance/change in time

b. find the vertex of this parabola using the method you learned.
From the vertex you can read off the maximum height and when that max happened.

c. The time shown in the vertex tells you that, from 0 to the time of the max will be the time
when the height increases, after that the height will be decreasing.

This graph should confirm your answer.
http://www.wolframalpha.com/input/?i=plot+h+%3D+98t+-+4.9t2

a. To find the average rate of change over the first 2 seconds, we need to calculate the change in height divided by the change in time.

Change in height = h(2) - h(0)
= (98(2) - 4.9(2)^2) - (98(0) - 4.9(0)^2)
= (196 - 19.6) - 0
= 176.4

Change in time = 2 - 0
= 2

Average rate of change = Change in height / Change in time
= 176.4 / 2
= 88.2 m/s

So, the average rate of change over the first 2 seconds is 88.2 m/s.

b. To find the point at which the rocket reaches its maximum height, we need to find the vertex of the parabolic equation h = 98t - 4.9t^2.

The equation h = 98t - 4.9t^2 is in the standard form of a quadratic equation: h = at^2 + bt + c, where a = -4.9, b = 98, and c = 0.

The t-coordinate of the vertex can be found using the formula t = -b / 2a.

t = -98 / (2 * -4.9)
= -98 / -9.8
= 10

To find the maximum height, substitute t = 10 into the equation h = 98t - 4.9t^2.

h = 98(10) - 4.9(10)^2
= 980 - 4.9(100)
= 980 - 490
= 490

So, the rocket reaches its maximum height of 490 meters at t = 10 seconds.

c. To determine the intervals during which the rocket's height is increasing and decreasing, we need to analyze the sign of the coefficient of the t^2 term.

Since the coefficient of the t^2 term is negative (-4.9), the quadratic function represents a parabola opening downwards. This means that the height is increasing when the rocket is initially launched and is decreasing after reaching its maximum height.

Therefore, the rocket's height is increasing from t = 0 seconds until it reaches its maximum height at t = 10 seconds, and it is decreasing afterwards.

To find the average rate of change over the first 2 seconds (a), we need to find the difference in the height of the rocket after 2 seconds and the height of the rocket at the beginning (t = 0), and divide it by the time interval.

Step 1: Find the height of the rocket after 2 seconds:
Plug in t = 2 into the equation for height:
h = 98(2) - 4.9(2^2)
= 196 - 4.9(4)
= 196 - 19.6
= 176.4 meters

Step 2: Find the height of the rocket at the beginning (t = 0):
Plug in t = 0 into the equation for height:
h = 98(0) - 4.9(0^2)
= 0 - 4.9(0)
= 0 - 0
= 0 meters

Step 3: Calculate the average rate of change over the first 2 seconds:
Average rate of change = (Change in height) / (Change in time)
Average rate of change = (176.4 - 0) / (2 - 0)
Average rate of change = 176.4 / 2
Average rate of change = 88.2 meters per second (m/s)

So, the average rate of change over the first 2 seconds is 88.2 m/s.

To find at what point the rocket reaches its maximum height (b), we can use both graphical and algebraic solutions.

Graphical Solution:
- Plot the graph of the height equation h = 98t - 4.9t^2.
- The maximum height of the rocket corresponds to the highest point on the graph.
- Identify the x-coordinate (t) of this point on the graph.

Algebraic Solution:
- The equation for the height of the rocket is h = 98t - 4.9t^2.
- We need to find the maximum value of h.
- The maximum point of a quadratic equation in the form ax^2 + bx + c can be found using the formula t = -b / (2a).
- In this case, a = -4.9 and b = 98. Plugging these values into the formula, we get t = -98 / (2 * -4.9).
- Simplifying, t = -98 / -9.8 = 10.
- The rocket reaches its maximum height at t = 10 seconds.

Therefore, the rocket reaches its maximum height at t = 10 seconds.

To determine the intervals when the rocket's height is increasing or decreasing (c), we can examine the coefficient of the t^2 term in the equation for height.

In the equation h = 98t - 4.9t^2, the t^2 term is negative (-4.9t^2). When the coefficient of the t^2 term is negative, the graph opens downwards. Thus, the height will be increasing when the coefficient of t^2 is negative.

Since the coefficient of t^2 is -4.9, the rocket's height will be increasing when:

-4.9t^2 < 0

To find the values of t for which this inequality is true, we solve for t:

t^2 > 0 (dividing both sides by -4.9)

Since t^2 is always greater than zero, the height is increasing for all values of t.

Therefore, the rocket's height is increasing over all intervals.