Section 1.1&1.2
Which is a function?
person --> birthday
birthday --> person
Person --> birthday is a function because every person (input) has one birthday (output), but one birth date can belong to multiple individuals
Are 2x+ 1 = 5 and x+ 1 = 3 are equivalent equations?
Yes, equivalent equations have the same solutions. x= 2
What is the constant of proportionality equation?
Q=kt
k=Q/t
where Q is the output and t is the input
What are the three terms for the m value, and what are the 3 terms for the b value in y=mx+b?
m= rate of change, slope, coefficient
b= initial value, y intercept, constant
Create expressions from descriptions of calculations:
Double x, subtract it from y
y-2x
The cost of a taxi ride as a function of the distance traveled (in miles) is given byC(d) = 2.5d. How far did you travel if the cost of your taxi ride was $32.50?
2.5d=32.50
d=13
The number of books you read in a semester is directly proportional to the number of classes you’re taking. When you’re taking 4 classes you read 24 books.
1. Find the constant of proportionality and give its units.
2. If you take 6 classes, how many books do you read?
1. k=Q/t ; 24 books/4 classes= 6 books
2. Q=kt; so (6)*(6 classes)= 36 books
Are the following expressions linear or not? If so, find the coefficient and constant term.
1. 2x^2+ 1
2. 2(x−1) + 3x
1. no, the x is not to the power of 1
2. yes, 5x-2 (5 is coefficent, -2 is constant)
Sales tax in Indiana is 7%. Write an expression for the sales tax on an item whose price is p dollars. Write an expression for the total cost of an item whose price is p dollars. If the price of the item were $20 what is the total cost?
Tax expression: .07p
Expression for $20 item: p+.07p
($20)+(.07)($20)= $21.40
We can express global sea levels, in meters, as a function of time, in years.Sea levels were at a low point about 22,000 years ago. Since then they’ve risen about120 meters. Find the average rate of change of sea levels during this time period.
120 meters/ 22,000 years= .0055 meters per year
The population of a town (t) years after it is founded is given by P(t) = 600t+ 450.
1. What was the population when the town was founded?
By how much does the population increase during the first year?
1. P (0)= 450 (looking for the initial value here)
2. P (1)= 600(1) + 450= 1050 people
If f(x) = 2.5x+ 4 is a linear function giving the cost of a taxi ride as a function of the distance traveled (in miles), then how far can you travel if you have $26.50?
$26.50=2.5x + 4
x= 9 miles
Use the graph of a function y = g(x) to answer the following questions
*Graph on board*
1. What is g(0)?
2. For which x-values is g(x) = 0?
3. What is the average rate of change of this function from x = 0 to x = 2?
1. 0
2. when x is: -3, 0, +3
3. 5/2
For each of the parts below, state whether or not the expressions are equivalent. If so, use the rules of algebra to turn one expression into the other. If not, find values for the variables showing they are not equivalent.
1. x(2x + 3) and 2x^2 + 3x
2. x^2 + 1 and (x + 1)^2
1. expressions are the same (distribute the x through)
2. expressions are not the same (ex: plug in 2 for x)
Let C(x) = 15x + 60 give the cost of your cell phone bill, in dollars, as a function of the number of gigabytes of data you’ve used.
1. What is the rate of change of this linear function? Interpret its meaning. (Be sure to include the units in your answer.)
2. What is the initial value of this linear function? Interpret its meaning. (Be sure to include the units in your answer.)
y=mx+b
m=slope ($15) which is our rate of change, b= y-intercept (0,60) which is our initial value (you pay $60 before using any data)
Write equations for the following linear function in slope-intercept form.
1. the linear function whose graph passes through the points (0, 4) and (1, 7)
y= 3x+4
Use the table to answer the following questions.
*Table on white board*
1. What is f(5)?
2. What is f(1)?
3. For which x values is f(x) = 1?
4. For which x values is f(x) = 3?
5. For which x values is f(x) ≥ 2?
1. 3
2. -1
3. x=2
4. x=3,5
5. x= 3, 4, 5
For each equation below, state whether or not y is directly proportional to x. If so, give the constant of proportionality
1. y=x/2
2. y = 2/x
3. y = 2x + 1
1. yes, (1/2 * x); k= 1/2
2. no, cannot be rearranged to equal Q=kt
3. no, the + 1 makes it not Q=kt
Let P(x) = 105+15(x−3) also give the cost of your cell phone bill, in dollars, as a function of the number of gigabytes of data you’ve used. (C(x) = 15x + 60)
1. Are C(x) and P(x) the same function? Explain your answer.
2. Interpret the meaning of the constants 3 and 105. (Be sure to include the units in your answer.)
1. C(x) 15x+60 and factored out P(x) also is 15x+60
2. When you use 3 gigs of data, the total bill is $105
Write equations for the following linear function in slope-intercept form.
1. the linear function whose graph has slope -2 and passes through the point (1, −2)
1. y= mx+b
f(x)= y0 + m (x - x0)
= -2+-2(x-1)
y= -2x