Unit Rates & Proportional Relationships (y = kx)
Question: A car travels 180 miles in 3 hours at a constant speed. What is the unit rate in miles per hour?
60 miles per hour.
Runner A's speed is modeled by y = 8x (where x is seconds and y is meters). Runner B runs 15 meters in 2 seconds. Who is running faster?
Runner A (8 m/s vs. Runner B at 7.5 m/s).
When finding the slope of a line on a coordinate plane, what does the "rise" represent?
The vertical change (change in y).
In the linear equation y = mx + b, what do the variables m and b represent
m represents the slope (rate of change) and b represents the y-intercept (starting value).
A plumber charges a flat service fee of 50 dollars plus 30 dollars per hour of work. What value represents the y-intercept in this scenario?
50 (the flat initial fee).
Write an equation in the form y = kx for a proportional relationship where 4 pounds of apples cost $10.
y = 2.5x (or $y = 2.50x).
wo lines represent proportional relationships on the same graph starting at (0,0). Graph A is steeper than Graph B. Which graph has a greater unit rate?
Graph A.
What is the slope of a line that goes up 6 units for every 2 units it moves to the right?
3 (calculated as 6 / 2).
What is the y-intercept of the line given by the equation y = 4x - 7?
-7 (or the point (0, -7)).
A gym membership costs 20 dollars per month plus a one-time registration fee of 35 dollars. Write an equation modeling total cost C after m months.
C = 20m + 35.
What specific coordinate point must the graph of every proportional relationship pass through?
The origin (0, 0).
Bakery A sells 6 cookies for 9 dollars. Bakery B sells cookies using the equation c = 1.25n, where c is total cost and n is the number of cookies. Which bakery charges less per cookie?
Bakery B (1.25 dollars per cookie vs. Bakery A at 1.50 dollars per cookie).
A slope triangle drawn on a line has a vertical height of 4 units and a horizontal base of 8 units. What is the slope of the line?
1/2 (or 0.5)
A line crosses the y-axis at (0, 4) and has a slope of -2. What is its equation in y = mx + b form?
y = -2x + 4.
A 12-inch candle burns down at a constant rate of 1.5 inches per hour. What is the slope in the equation representing the candle's height over time?
-1.5 (or negative 1.5 inches per hour).
If the coordinate point (1, 7) lies on the graph of a proportional relationship, what is the constant of proportionality k?
k = 7
Line A passes through (0,0) and (4, 12). Line B passes through (0,0) and (5, 20). Which line has a larger constant of proportionality, and what is its value?
Line B, with a constant of proportionality of 4 (compared to Line A's constant of 3).
Why do any two slope triangles drawn on the same straight line give the exact same slope ratio?
Because the slope triangles are similar triangles, meaning their corresponding side ratios are equivalent.
How does the graph of y = 3x + 5 compare visually to the graph of the proportional line y = 3x?
It is parallel to y = 3x, but shifted up by 5 units on the vertical axis.
Service A costs 10 dollars per month with no setup fee. Service B costs 5 dollars per month plus a 15 dollar setup fee. After how many months will both services cost the exact same amount?
3 months (solving 10m = 5m + 15).
A table shows that 3 bags of soil cover 12 square feet, and 7 bags cover 28 square feet. What equation represents the total area y covered by x bags of soil?
y = 4x.
Maya fills a pool at a rate of 15 gallons per minute. Sam fills his pool according to g = 18m. After 10 minutes, how many more gallons of water are in Sam's pool than Maya's?
30 gallons more (180 gallons vs. 150 gallons).
Find the slope of the line passing through the points (2, 5) and (6, 17).
(calculated as (17 - 5) / (6 - 2) = 12 / 4).
A line passes through (0, 10) and (3, 16). What is the equation of the line in slope-intercept form?
y = 2x + 10.
A phone plan table shows the total monthly cost y for x gigabytes used: (1, 25 dollars), (2, 35 dollars), (3, 45 dollars). What is the base monthly fee before any data usage?
5 dollars (since data costs 10 dollars per GB, subtracting 10 dollars from 25 dollars gives a base fee of 15 dollars).