Rate of Change
A car travels 180 miles in 3 hours. What is its rate of change?
60 miles per hour
A store sells 6 notebooks for $12. What is the unit price?
$2 per notebook
The equation
[ y=6x ]
represents a proportional relationship. What is the constant of proportionality?
Strategy: In (y=kx), k is the constant of proportionality.
6
Which coordinate represents a point that is 3 units right and 5 units up from the origin?
Strategy: Right = positive x. Up = positive y.
(3,5)
If a problem says "per", what mathematical operation should you usually consider?
Strategy: "Per" means for each, which usually signals a unit rate or division.
Division
A table shows:
x y
1 5
2 10
3 15
What is the rate of change?
Strategy: Find how much y changes when x increases by 1.
[ 10-5=5 ]
5
A car travels 240 miles using 8 gallons of gas. What is the unit rate?
30 miles per gallon
A table shows:
x y
1 4
2 8
3 12
4 16
What is the constant of proportionality?
4
What is the slope of a line passing through ((1,2)) and ((3,8))?
3
Which is an equation?
A. (4x+7)
B. (4x+7=19)
C. (7x) D. (3+8)
An equation contains an equals sign.
Correct Answer: B. (4x+7=19)
A line passes through the points (2, 7) and (5, 16). What is the rate of change?
3
A store advertisement says:
3 shirts for $45
What is the cost per shirt?
$15 per shirt
A recipe uses 3 cups of flour for every 2 batches of cookies. How many cups of flour are needed for 5 batches?
Strategy: First find the constant of proportionality
7.5 cups
A line passes through ((0,4)) and ((2,10)). What is the y-intercept?
Strategy: The y-intercept occurs when x = 0.
The point ((0,4)) is already on the y-axis.
Answer: 4
Simplify:
3(x+5)
Strategy: Multiply 3 by every term inside the parentheses.
3(x)+3(5)
Correct Answer:
3x+15
A line has a rate of change of 4 and a y-intercept of 2. Which equation represents the line?
A. (y=2x+4)
B. (y=4x+2)
C. (y=4x-2)
D. (y=2x-4)
Answer: B. (y=4x+2)
A runner runs 2½ miles in ½ hour.
What is the runner’s unit rate in miles per hour?
A. 2 miles per hour
B. 3 miles per hour
C. 4 miles per hour
D. 5 miles per hour
Answer: D. 5 miles per hour
The relationship between (x) and (y) is shown below:
x y
2 10
4 20
6 30
What is the equation?
[y=5x]
A proportional relationship contains the point ((4,20)). What is the constant of proportionality?
5
A student writes:
4(x+3)=4x+3
Is the student correct? Explain
No.
Strategy: Use the distributive property. The 4 must multiply both terms.
4(x+3)=4x+12
4x+12
A taxi charges a $5 starting fee plus $2 for every mile traveled. What is the rate of change and what does the y-intercept represent?
Strategy: The amount added for every mile is the rate of change. The amount charged before any miles are traveled is the y-intercept.
Which is the better deal?
Strategy: Find the unit price for each.
Option B, because $0.45 per ounce is cheaper.
Two students disagree.
Student A: The constant of proportionality is 3 because (y) increases by 3.
Student B: The constant of proportionality is 4 because (y\div x=4).
Who is correct?
x y
1 4
2 8
3 12
Student B. The constant of proportionality is 4.
A line passes through ((0,0)) and ((6,18)). Write the equation and identify the constant of proportionality.
3
A school club sells T-shirts. The table shows:
Shirts Sold Money Earned
2 $30
4 $60
6 $90
8 $120
What is the unit rate, constant of proportionality, and equation?