Linear Functions
The 1st, 3rd, and 5th values of a table are 7,17, and 27.
What are the 2nd and 4th terms.
What is the explicit equation?
12 and 22
y=5x+2
A colony of bacteria starts with 100 cells and doubles every hour. a) Write an exponential equation to represent the number of bacteria (B) after h hours. b) How many bacteria will there be after 12 hours?
B=100(2)h
409,600
The student council is selling raffle tickets. They start with $200 and earn $25 per ticket sold. a) Write a linear equation for the total money (M) after selling n tickets b) Is this a discrete or continuous model? Explain why. c) Would it make sense to calculate the money earned after selling 5.7 tickets?
M=25n+200
Discrete
No
I have the slope -7 and the point (5,8). What is the Point Slope Form?
-7(x-5)+8
A new content creator tracks their followers:
a) Is this growth linear or exponential? Prove it mathematically. b) Write an equation to model the number of followers (F) after w weeks. c) Is this a discrete or continuous model? Explain why. d) Predict the number of followers in Week 6.
Exponential,r=3
F=50(3)w-1
Discrete
12150
A concert venue charges $45 for the first row seats and each row behind it decreases in price by $3. Write a linear equation to find the price (P) for any row number (r).
y=-3x+48
A new car worth $32,000 depreciates by 15% each year. a) Write an exponential equation to find the car's value (V) after t years. b) What will be the car's value after 4 years?
V=32000(.85)t
$16,704.20
A bacteria culture starts with 50 cells and doubles every hour. a) Write an exponential equation for the number of cells (C) after t hours b) Is this model discrete or continuous? Explain why.
C=50(2)t
Discrete
I have the points (-3,4),(2,8),(7,12), and (12,16).
What is an equation for point slope form?
4/5(x+3)+4
4/5(x-2)+8
4/5(x-7)+12
4/5(x-12)+16
An electric car's battery charge decreases by 30 miles for each hour of highway driving. Starting with a full charge of 300 miles:
a) Is this situation linear or exponential? Justify your answer. b) Write an equation for remaining range (R) after h hours of driving. c) Is this a discrete or continuous model? Why? d) Would it make sense to calculate the range after 2.25 hours? Explain.
Linear
R=-30h+300
Continuous
Yes
The school library starts with 2,500 books and adds 150 new books each year. a) Write a linear equation to represent the total number of books (B) after y years. b) How many books will the library have after 6 years?
y=150x+2500
3400
At year 0 I have $500 that grows to $600 in year 1 and then $720 in year 2.
What is the common ratio?
What is the explicit equation for the amount (A) after t years?
r=1.2
A=500(1.2)t
A hot drink starts at 90°C and cools by approximately 8% per minute in room temperature. a) Write an exponential equation for the temperature (T) after m minutes b) Is this a discrete or continuous model? Explain why. c) Can you find the temperature at any point in time (including partial minutes)?
T=90(.92)m
Continuous
Yes
I have the equation y=-8(x-10)-30 in point slope form.
What is the slope?
What is the point on the line?
What is the equation in slope intercept form?
Slope=-8
Point:(10,-30)
y=-8x+50
A house purchased for $200,000 either:
a) Write equations for both scenarios where V = value and t = years b) Which scenario is linear and which is exponential? c) After 10 years, which scenario results in a higher value?
V=200000+12000t
V=200000(1.06)t
Linear and Exponential
Exponential (358,169.53 vs 320,000)
James runs at a constant pace. He passes the 400m mark at 2 minutes and the 1000m mark at 5 minutes. a) Write a linear equation to represent his distance (d) in meters after t minutes. b) Using your equation, find how long it will take James to run 1600 meters.
d=200t
8 minutes
A viral post starts with 40 views and triples every 1 hour. a) Write an exponential equation to represent the number of views (v) after h hours b) How many views will the post have after 16 hours?
v=40(3)h
1,721,868,840
A theater charges $12 per ticket and has fixed daily costs of $400. a) Write a linear equation for the daily profit (P) based on number of tickets (n) sold b) Is this a discrete or continuous model? Explain why. c) If they sold 108 tickets, what was their profit? d) Would it make sense to calculate profit for 108.5 tickets?
P=12n-400
Discrete
$896
No
I have the points (-3,-6) and (-1,-9).
What is the equation in point slope form?
-3/2(x+3)-6
-3/2(x+1)-9
In a lab experiment, a chemical solution either:
Starting with 200 mL:
a) Write equations for both cases where V = volume and h = hours b) Identify which case is linear and which is exponential c) Is the volume a discrete or continuous measurement? Explain.
V=200-15h
V=200(.85)h
Linear and Exponential
Continuous
A scientist records temperatures every hour. The temperatures form an arithmetic sequence: 23.5°C, 22.8°C, 22.1°C, ... a) Find the common difference b) Write an explicit equation for the temperature (T) at the nth hour c) What will be the temperature at the 8th hour?
d=-0.7
T=-0.7n+24.2
18.6 C
A radioactive sample has an initial mass of 80 grams. After 1 year, 60 grams remain. a) Find the common ratio b) Write an exponential equation for the remaining mass (M) after t years c) How much will remain after 4 years?
r=0.75
M=80(0.75)t
25.3125 grams
A freshly brewed cup of coffee starts at 85°C and cools at approximately 12% per minute when left in a room at 22°C.
a) Write an exponential equation for the coffee's temperature (T) after m minutes
b) Is this a discrete or continuous model? Explain why.
c) Can you calculate the temperature at any point in time (like 2.75 minutes)? Why or why not?
T=85(.88)m
Continuous
Yes
I have the points (-10,4) and (-9,12).
What is the equation in slope intercept form?
y=8x+84
Consider two different library growth scenarios starting with 5,000 books:
a) Write equations for both scenarios where B = books and y = years b) Classify each as linear or exponential and explain how you know c) Is the number of books discrete or continuous? Why? d) Calculate the difference in total books between the two scenarios after 5 years e) Which growth model would result in more books after 10 years?
B=5000+400y
B=5000(1.08)y
Linear and Exponential
Discrete
Linear: 7000, Exponential: 7346
Linear: 9000, Exponential: 10794