Slope & Graphs
What is the slope of a line with the equation y = 4x − 7?
m = 4 (the coefficient of x in y = mx + b)
Solve for x: x + 9 = 15
x = 6 (subtract 9 from both sides)
State the Pythagorean theorem.
a² + b² = c² where c is the hypotenuse (longest side, opposite the right angle).
What is 2⁵?
32 (2 × 2 × 2 × 2 × 2 = 32)
A line passes through (0, 3) and (2, 7). What is the slope?
rise ÷ run = (7−3) ÷ (2−0) = 4 ÷ 2 = 2
Solve for x: 4x − 3 = 17
4x = 20 → x = 5 (add 3, then divide by 4)
A right triangle has legs of 6 and 8. What is the hypotenuse?
6² + 8² = 36 + 64 = 100 = c² → c = 10
A population of 50 bacteria doubles every hour. How many are there after 3 hours?
50 × 2³ = 50 × 8 = 400 bacteria
Your phone battery is at 100% and loses 8% per hour. Write the equation for battery (y) after x hours. (multiple possible answers)
y = −8x + 100 (slope = −8, y-intercept = 100)
You buy 3 identical notebooks and a pen for $2. You spend $11 total. How much does one notebook cost? Set up and solve.
3n + 2 = 11 → 3n = 9 → n = $3 per notebook
A ladder 13 feet long rests against a wall. The base is 5 feet from the wall. How high up the wall does it reach?
5² + b² = 13² → 25 + b² = 169 → b² = 144 → b = 12 feet
What is the exponential growth curve (sometimes called the j-curve) and why does it fool people?
The J-curve is the shape of exponential growth: flat for a long time, then sudden explosion. People see the flat part and assume nothing is happening — then the steep part catches them off guard.
A line has slope 3 and passes through the point (2, 9). What is its y-intercept? Write the full equation.
y = 3x + b → 9 = 3(2) + b → 9 = 6 + b → b = 3. Full equation: y = 3x+3
Explain in words what it means to 'solve for x' — and why you must do the same operation to both sides.
Solving for x = finding the value that makes the equation true. Both sides must stay equal (balance scale) — if you change one side, you must change the other identically.
Two points on a graph: (1, 2) and (4, 6). Find the distance between them using the Pythagorean theorem.
Δx = 3, Δy = 4. d = √(3² + 4²) = √(9+16) = √25 = 5
$100 in a savings account doubles every 10 years (2× growth). How much after 40 years? How does that compare to simple interest adding $100 every 10 years?
Exponential: $100 × 2⁴ = $1,600. Simple interest: $100 × 5 = $500. Exponential is more than 3× bigger.
A car travels at constant speed. At t = 2 hours it has gone 90 miles. At t = 5 hours it has gone 225 miles. Write the equation for distance traveled and find the car's speed.
m = (225−90) ÷ (5−2) = 135 ÷ 3 = 45 mph. y-intercept: 90 = 45(2) + b → b = 0. Equation: y = 45x. The car started from the origin — it was already moving at t = 0.
The number trick: pick n, multiply by 3, add 12, divide by 3, subtract n. Prove algebraically what you always get. Then modify the trick so you always get 7 instead.
n → 3n → 3n+12 → n+4 → 4. You always get 4. To get 7: change "add 12" to "add 21" → 3n+21 → n+7 → 7.
A right triangle has legs of length 20 and 21. What is the hypotenuse? No calculator.
20² + 21² = 400 + 441 = 841 = 29². Hypotenuse = 29. (20-21-29 is a Pythagorean triple — just like 3-4-5 and 5-12-13.)
A population of bacteria starts at 1. It doubles every hour. How many bacteria are there after 8 hours? Write the equation and solve without a calculator.
y = 1 · 2^8 = 2^8. Build it up: 2→4→8→16→32→64→128→256. After 8 hours: 256 bacteria.