1) What is the meaning and purpose of the axis of a quadratic function? 2) Solve the inequalities

Please see the attached file for detailed solutions.

) What is the meaning and purpose of the axis of a quadratic function?
For a quadratic function of , it can also be written as

Then line of axis is , this is also the symmetric line of the graph of this quadratic function.
When you graph the curve, you can just draw the left or right side of the , then the other half can be obtained by the symmetry.

Another use of axis of the quadratic function is the intersection of the symmetric line with the quadratic curve is the vertex of the quadratic function. That is the vertex is .

2) Solve the inequalities. State the solution set using interval notation for:

(x - 4) / (x + 3) < 0

case 1

Case 2:

In interval notation:

and then for:

(x - 4) / (x + 3) > 0

case 1

Case 2:

So the solution is x<-3 or x>4.
Interval notation:

What do you get when uniting the two sets of solutions?
If you unite and , you get

Or

3) The surface area A, of a cylinder with height h, and radius r, is
given by the equation A = 2prh + 2pr^2. Find the radius of a cylinder
with height 2 cm and area 4p cm2.

Plug into the above equations

Using the quadratic formula,

The radius has to be greater than 0. so the radius is cm.

4) An arrow is shot straight upward with a velocity of 96 feet per second at instant t0 = 0; its altitude is defined in time by the following function:

f(t) = -16t² + 96t + 6.
From what altitude is it shot? For how many seconds is this arrow more than 86 feet high?
When t=0
, so the arrow was shot at altitude of 6 feet.

(a)
Note: Remember to change the sign of the inequality when multiplying or dividing a negative number.

Case 1

Case 2

So the solution to the inequality (a) is

That is , when time is greater than 1 second and less than 5 seconds, the altitude is higher than 86 feet.