Exponential Growth and Decay
A bacteria culture starts with 500 cells and grows by 12% each hour. How many cells will there be after 5 hours?
y=500(1.12)5
881 Cells
Find the domain and range.
f(x)=3x-5
Domain: All real number (-∞,∞)
Range: All real numbers (-∞,∞)
(x+3)(x+5)
x(x+5)+3(x+5)
=x2+5x+3x+15
=x2+8x+15
answer: x2+8x+15
find the degree, leading coefficient, and end behavior.
f(x)=3x4−2x+1
Degree: 4 (even)
Leading coefficient: positive
End behavior:
x→−∞, y→∞
x→∞, y→∞
A set of test scores is normally distributed with a mean of 80 and a standard deviation of 5. What percent of students scored between 75 and 85?
75 and 85 are 1 standard deviation from the mean. Empirical Rule - 68%
Answer: 68%
A radioactive material loses 8% of its mass each year. If you start with 200 grams, how much remains after 10 years?
y=200(0.92)10
86.8 grams
Find the domain and range.
f(x)=2/x-4
Domain: all real numbers except x=4
(-∞,4)∪(4,∞)
Range: all real numbers except y=0
(−∞,0)∪(0,∞)
(2x−4)(x+6)
2x(x+6)−4(x+6)
=2x2+12x−4x−24
=2x2+8x−24
answer: 2x2+8x−24
find the degree, leading coefficient, and end behavior.
g(x)=−5x3+7x−9
Degree: 3 (odd)
Leading coefficient: negative
End behavior:
x→−∞, y→∞
x→∞, y→−∞
Heights of adult men are normally distributed with mean 70 in and standard deviation 3 in. What percent of men are between 64 and 76 inches?
64 and 76 are 2 standard deviations from the mean. Empirical Rule - 95%
Answer: 95%
You invest $1,000 in an account that grows at 5% per year, compounded annually. How much will you have after 15 years?
A=1000(1.05)15
$2078.93
Find the domain and range.
f(x)=square route of x+1
Domain: x+1≥0⇒x≥−1
[−1,∞)
Range: y≥0
[0,∞)
(x−7)(x−2)
x(x−2)−7(x−2)
=x2−2x−7x+14
=x2−9x+14
Answer: x2−9x+14
find the degree, leading coefficient, and end behavior.
h(x)=2x5−x2+4
Degree: 5 (odd)
Leading coefficient: positive
End behavior:
x→−∞, y→−∞
x→∞, y→∞
Find the z‑score for a value of 92, given mean 80 and standard deviation 4.
z=92−80/4=12/4=3
Answer: z = 3
A car is worth $25,000 when new and loses 15% of its value each year. What is its value after 6 years?
v=25000(0.85)6
$9425
Find the domain and range.
f(x)=1/x2
Domain: all real numbers except x=0
(−∞,0)∪(0,∞)
Range: y>0
(0,∞)
(3x+1)(x2−4)
3x(x2−4)+1(x2−4)
=3x3−12x+x2−4
=3x3+x2−12x−4
answer: 3x3+x2−12x−4
find the degree, leading coefficient, and end behavior.
p(x)=−4x2+6x+10
Degree: 2 (even)
Leading coefficient: negative
End behavior:
x→−∞, y→−∞
x→∞, y→−∞
A student scored 72 on a test where the mean was 80 and the standard deviation was 8. Find the z‑score.
z=72−80/8=−8/8=−1
Answer: z = –1
A small town has a population of 8,000 and grows at 3% per year. What will the population be in 12 years?
p=8000(1.03)12
11406 people
Find the domain and range.
f(x)=x2−4
Domain: all real numbers (−∞,∞)
Range: parabola opens up, minimum at −4
[−4,∞)
(x+4)(x2+2x+3)
x(x2+2x+3)+4(x2+2x+3)
=x3+2x2+3x+4x2+8x+12
combine like terms: =x3+6x2+11x+12
answer: x3+6x2+11x+12
find the degree, leading coefficient, and end behavior.
k(x)=x7−3x+2
Degree: 7 (odd)
Leading coefficient: positive
End behavior:
x→−∞, y→−∞
x→∞, y→∞
A distribution has mean 50 and standard deviation 10. What percent of data lies between 30 and 70?
30 and 70 are 2 standard deviations from the mean. Empirical Rule → 95%
Answer: 95%