A few Math questions!(: 10 POINTS BEST AWNSER!?

9. What is the vertex form of the equation?
y = x^2 – 6x + 8 (2 points)
13. What is the expression in factored form?
6x^2 + 4x (2 points)
14. What is the expression in factored form?
4x^2 + 11x + 6 (2 points)

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What does interpret the value of the average rate of change mean?

This is my problem and my math book isn't giving me any information about what this question means..
"The graph shows the population P in a small industrial city from 1950-2000. The variable X represents the number of years since 1950."
Graph below points are on X = 20,25,40,50 ; Y= 40,50,40,20
50-
40-
30-
20-
10-
0 ____________________________
--------------10 20 30 40 50
(a) Determine the intervals on which the function P is increasing and on which it is decreasing.
My answer= Increasing [0,20] [20,25] ; Decreasing [25,40] [40,50]
(b) What was the average rate of change of P between X=20 and X=40 ?
My answer = 40 - 40 / 40-20 = 0/20 = 0
(c) Interpret the value of the average rate of change that you found in part (b)
** This is where I am lost. I'm not sure what the question is actually asking, I've done some searching on the internet about the value of the average rate of change, although I think I'm either finding the wrong thing, or not quite catching the concept. Intuitively I consider the fluctuation between the two intervals, but as the past has shown me, intuition and math don't necessarily make good partners ;X

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Why is the answer 2y/x^2 and not 2x^-2*y?

x^(-a) = 1/x^a
x^(-2) = 1/x^2
Therefore, 2y/x^2 and 2x^(-2)y are equal
But often, you will be asked to use positive exponents,
so it's best to answer 2y/x^2

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How do you solve 7+3t over 4 =-t over 8?

This is: ?
(7+3t)/4 = -t/8
If it was the equation above do the following..
multiply both sides with 8 which gives you:
2(7+3t) = -t (get 2 due to (8/4 = 2))
14 + 6t = -t
6t + t = -14
t = -2

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Describe the effect of each number on the graph of f(x) =7 (-1x + 1)^3 +4.?

Describe the effect of each number on the graph of f(x) =7 (-1x + 1)^3 +4.
Match:
7
1
4
-1
3
Match with:
a. shifts the graph up
b. identifies the common graph to be used
c. shifts the graph to the left
d. stretch
e. reflects the graph over the y axis

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differentiation of exponential functions?

you have fraction, note that
[a/b]' = (a'b-b'a)/b^2
in your case a=x
b=e^x
as you know
x'=1
(e^x)' = e^x
so we get
(1*e^x - xe^x)/e^(2x)

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Find all the zeros of each equation.?

1. x^5 – 3x^4 – 15x^3 + 45x^2 – 16x + 48 = 0
A. 3, 4, i
B. 4, –4, i,–i
C, 3, 4, –4, i, –i
D. 3, –4, –i
2. x^3 – x^2 + 5x – 5 = 0
A. i, –i, 1
B. i, –i, –1
C. , –, –1
D. i, 1
3. 7x^2 – 144 = –x^4
A. 3, –3, 4i, 0
B. 3, 4i
C. 3, –3, 4i, –4i
D. –3, –4i

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Given that x²+y²=14x+6y+6,find the largest possible value of the expression E=3x+4y?

x^2-14x +y^2-6y=6
(x-7)^2 +(y-3)^2 = 6+49+9 = 64 = 8^2
this is a circle centre (7,3) radius 8
y = 3 + 8 sin[theta]
x= 7+ 8 cos[theta]
E= 3(7+8cos( theta)) +4 (3 +8 sin[theta])
= 21 +24 cos (theta) + 12 +32 sin (theta)
= 33 +24 cos (theta)+ 32 sin(theta)
dE/ d (theta)= -24sin(theta) +32cos (theta)=0
32cos theta=24sin (theta)
tan (theta) = 32/24 = 4/3
theta = arctan(4/3)
E= 33 +24 cos (arctan(4/3)) + 32 sin(arctan(4/3)
=73

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For a cubic graph, how do you know if it goes below the x axis or above the x axis?

every cubic (or in general, every polynomial of odd order) crosses x-axis.
to find where that happens, you solve for y=0.
for example if you have cubic, it would be ax^3+bx^2+cx+d=0

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Quick! Do arithmetic sequences have to be addition and subtraction?

I am learning about arithmetic sequences, and my book only shows addition and subtraction. Are multipication patterns arithmetic sequences too?
Examples:
I know 2, 4, 6, 8, 10 is an arithmetic sequence. +2
But is 1, 3, 9, 27 an arithmetic sequence too? x3

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