SLOPE & PROPORTIONS
A proportional relationship passes through (0,0) and (4,12). What is the unit rate?
What is 3?
The number of solutions to the equation 5x−3=2x+9
Answer: What is one solution (x=4)?
When two lines intersect at exactly one point, the system has this type of solution.
What is one solution?
A function whose graph is a straight line and can be written as y=mx+b
Answer: What is a linear function?
In the function C(t)=50+25t representing cost over time, this number represents the initial value.
Answer: What is 50?
The slope between any two points on a non-vertical line is always this.
What is the same (or constant)?
Solve: 3(x+4)=2x−1
Answer: What is x=−13?
Solve the system:
x+y=7
x−y=3
Answer: What is (5,2)?
Function A: y=2x+5. Function B: A line through (0, 3) with slope 4. Which has the greater y-intercept?
Answer: What is Function A (y-intercept = 5)?
A gym membership costs $30 to join plus $15 per month. Write the linear function for total cost after m months.
What is C(m)=30+15m
y= 15x+30
A proportional relationship has these two key characteristics on a graph.
What are "passes through the origin (0,0)" and "forms a straight line"?
This answer to this equation 4x+7=4x+12.
What is no solution?
Use substitution to solve:
y=3x−2
2x+y=8
Answer: What is (2,4)?
Function S: y=4x−1. Function R is shown in this table. At what x-value do both functions have the same output?
x y
1 7
2 9
3 11
4 13
What is x=3? (Both functions equal 11 when x=3)
A candle starts at 8 inches tall and burns 0.5 inches per hour. When will it be completely burned?
Answer: What is 16 hours? (Function: h(t)=8−0.5t, solve h(t)=0
Compare these relationships:
Graph A shows a line through (0,0) and (5,15).
Table B shows: (0,0), (2,10), (4,20). Which has the greater unit rate?
What is Table B? (Table B: rate = 5, Graph A: rate = 3)
A student solved 2(3x−1)+4=6x+2 and got "2=2" as their final step. What does this mean about the solution to the equation?
What is "the equation has infinitely many solutions (or is an identity)" because the variables canceled out and left a true statement?
At a school carnival, adult tickets cost $12 and student tickets cost $8. If 150 tickets were sold for a total of $1,400, how many adult tickets were sold?
What is 50 adult tickets?
System:
a+s=150
12a+8s=1400
A function is described as "starting at -2 and increasing by 0.75 for each unit." Compare this to the function shown in the graph that passes through points (0, 1) and (4, 7). Which has the steeper rate of change? Greater=steeper
Answer: What is the graphed function? (Described function: rate = 0.75; Graphed function: rate=1.5; since 1.5 > 0.75, the graphed function is steeper)
The temperature at 6 AM was 45°F and rises 3°F each hour. At what time will it reach 75°F?
Answer: What is 4 PM (10 hours later)?
Function: T(h)=45+3h, solve T(h)=75
A recipe calls for 3 cups of flour for every 2 cups of sugar. If this relationship is graphed with sugar on the x-axis and flour on the y-axis, write the equation and explain what the slope represents.
Answer: What is y=3/2x or y=1.5x,
where the slope represents 1.5 cups of flour per cup of sugar?
Create an example of a linear equation with infinitely many solutions and explain why it works.
Answer: What is any equation like
3x+6=3(x+2) because both sides are algebraically identical (accept equivalent examples)?
Two streaming services offer different plans.
Service A: $15/month + $3 per premium movie. Service B: $25/month + $1 per premium movie.
For what number of premium movies per month would both services cost exactly the same? Set up and solve the system.
What is 5 premium movies?
(System: 15+3m=25+1m, solve to get 2m=10, so m=5. Both services cost $30 when you watch 5 premium movies.)
A linear function starts at 25 when x=0 and increases by 1.5 for each unit increase in x.
Compare its rate of change to the function f(x)=30−2x
What is "the first function is f(x)=25+1.5x with rate of change 1.5; the second has rate of change -2; they change in opposite directions"
A water tank holds 600 gallons and drains at 20 gallons per minute. A backup tank starts with 100 gallons and fills at 10 gallons per minute. Write functions for both tanks and find when the main tank will have twice as much water as the backup tank.
What is Main tank: M(t)=600−20t
Backup tank: B(t)=100+10t and
M(t)=2B(t) when
600−20t=2(100+10t)
which gives t=10?
(At 10 minutes: Main = 400 gallons, Backup = 200 gallons)