Equation Forms
Write the slope-intercept form of a line with slope 2 and y-intercept –3.
y=2x−3
A parking garage charges a $6 flat fee plus $1.50 per hour. Write an equation for cost after h hours.
C=6+1.5h
Identify a and b in f(x)=5(2^x).
a=5, b=2
In a real-world context, what does slope usually represent?
Rate of change (e.g., cost per unit, speed, growth per time)
Evaluate f(x)=5(2^x) at x=3.
40
Convert 3x-2y=10 to slope-intercept form.
y=(3/2)x−5
A taxi charges $4 flat fee plus $2.25 per mile. Write an equation for the cost after m miles.
C=4+2.25m
If b>1 in f(x)=a(b^x), does the situation represent growth or decay?
Growth
What does the y-intercept represent?
The starting value / initial condition
Evaluate f(x)=200(0.9^x) at x=5.
About 118.1
Write the equation of a line passing through (−1,2)(-1,2)(−1,2) with slope –3.
y−2=−3(x+1) or equivalent
Concert tickets cost $12 each, but drop to $10 each if more than 600 are sold. Is this an equation, inequality, or piecewise?
Piecewise function
A medicine starts at 300 mg and loses 40% per hour. Write the equation.
M(t)=300(0.6^t)
In f(x)=200(0.9^x), what does the 0.9 mean?
10% decay each step (keeping 90% each time)
Which grows faster long-term: Linear growth of +200 per year or exponential doubling every 10 years?
Exponential doubling
Rewrite y=(1/2)x−9 in standard form.
x−2y=18
Skating Rink A charges $8 + $5/hour. Rink B charges $10/hour. After how many hours is Rink B cheaper?
After 5 or more hours
A town has a population of 1,200 and grows by 8% per year. Write the equation.
P(t)=1200(1.08^t)
When is slope-intercept form easier to use than point-slope form?
When you know slope and y-intercept, or want to quickly graph
A gym charges $40 sign-up fee + $20/month. Write the equation and find cost after 9 months.
C=40+20m, C=220 at 9 months
Two points (2,5) and (4,9) are given. Write the equation of the line in any form.
Slope = 2 → y−5=2(x−2) or equivalent
A car rental is $25/day + $60 insurance, or $40/day with no insurance. For what number of days is Company A cheaper?
For fewer than 4 days
Option A: Start with $100, add $20 each year. Option B: Start with $100, grow by 15% each year. After 15 years, which is greater?
Option B (exponential growth passes linear growth)
How can you tell from a graph if a situation is exponential instead of linear?
Exponential curves grow/decay multiplicatively (curved), linear grows additively (straight line)
How do you decide if a situation should be modeled by an equation, inequality, or piecewise?
Equation = one rule; Inequality = includes limits/constraints; Piecewise = two or more rules depending on conditions