Exponents and Exponential Functions
Simplify the following. Write your answer using only positive exponents.
5-2x5y-2x-3
(x²)/(25y²)
Determine if the following sequence is arithmetic, geometric, or neither.
1, 3, 7, 11, 16,...
Neither!
Add:
(2x² -5x +7) + (13x² -11x -10)
15x² -16x -3
Factor:
x2 + 6x +8
(x+2)(x+4)
Is the parabola concave up or concave down?
y = 2x - 3x² + 2
Concave down
Simplify the following:
(2x5)4(3x2)3
(2x5)4(3x2)3 = 24(x5)433(x2)3 = 16x20*27x6 = 432x26
Determine if the following is arithmetic, geometric, or neither.
3, 1, -1, -3, ...
Arithmetic!
Common difference = -2
Subtract:
(3x² +2) - (4x² -x +5)
-x² +x -3
Solve:
3x² - 48 = 0
3x² = 48
x² = 16
x = -4, 4
What is the axis of symmetry for the parabola?
y = x² + 9
x = (opp b)/(2a) = 0/(2*1) = 0
x = 0
Evaluate the following exponential function at x = 3
f(x) = 2 * 4x
f(3) = 2* 43
f(3) = 2 * 64
f(3) = 128
What is the common ratio for the geometric sequence below?
2, -4, 8, -16, ...
r = -2
Multiply:
2x(2x - 5)
4x² -10x
Solve:
x² - 10x + 24 = 0
x = 4, 6
y = x² + 16x + 64
x = (opp b) / (2a) = -16/(2*1) = -8
y = (-8)² +16(-8) + 64 = 0
Vertex: (-8, 0)
Is the given exponential function an example of exponential growth or exponential decay?
f(x) = (1/2)x
Decay!
Write a formula for the following sequence.
27, 9, 3, 1, ...
A(n) = 27*(1/3)(n-1)
Multiply:
(2x+3)(x² -3x + 4)
2x³ -3x² -x+12
Solve:
3x² - x - 2 = 0
3x² - 3x + 2x - 2 = 0
(3x + 2)(x - 1) = 0
x = -2/3, 1
An object is launched directly upward at 64 feet per second (ft/s) from a platform 80 feet high. The function h(t) = -16t² + 64t + 80 gives the height of the object after t seconds. What will be the object's maximum height?
We want the y-coordinate of the vertex!
x = (opp b)/(2a) = -64/(2*-16) = -64/-32 = 2
y = h(2) = -16(2)² +64(2) + 80 = 144 feet
Suppose your parents deposited $6000 in an account paying 4.9% interest compounded quarterly. What will the account balance be after 18 years?
P = A(1 + r/m)m*t
P = 6000(1 + .049/4)12*4
P = 14416.90
What is the 10th term in the following sequence?
1, -3/2, 9/4, -27/8...
A(n) = 1*(-3/2)(n-1)
A(10) = 1*(3/2)9 = -19683/512
A circular mirror has a radius of 7x. The mirror has a square frame with side length 21x. What is the area of the frame?
Area of square = (21x)² = 441x²
Area of circle = π *(7x)² = 49x² *π
Area of frame =441x² - 49x²*π = 49x² (9-π)
Solve:
4x³ + 2x² - 2x = 0
2x(2x² + x - 1)
2x(2x² + 2x - x - 1)
2x(x + 1)(2x - 1)
x = 0, -1, 1/2
An object is launched directly upward at 64 feet per second (ft/s) from a platform 80 feet high. The function h(t) = -16t² + 64t + 80 gives the height of the object after t seconds. When will the object hit the ground?
Solve -16t² + 64t + 80 = 0!
16(-t² + 4t + 5) = 0
16(-t² -t + 5t + 5) = 0
16(-t +5)(t + 1) = 0
t = -1, 5 —> 5 seconds!