Week 31: Variation, growth, and inequalities on the grid

Field Book

113 to 116

Due

Test 28 due

Method

Paper first, every step

Variation is a way of saying two quantities are locked together. In direct variation they rise and fall together. In inverse variation one goes up exactly as the other comes down, and their product never changes. Exponential growth is what happens when something multiplies instead of adds, and in this class you compute it by hand.

What you have to be able to do

Set up and use direct and inverse variation
Compute exponential growth by hand
Graph a linear inequality and simplify a quotient of square roots

The procedures

Direct and inverse variation

Direct variation is y = kx. Both move the same direction, and the ratio stays constant.
Inverse variation is xy = k. They move opposite directions, and the product stays constant.
Either way, use the pair of values you are given to find k first, then answer the question with that k.

Graphing a linear inequality

Graph the boundary line as if the inequality sign were an equals sign.
Dashed line if the sign is strictly less or greater. Solid line if it includes equals.
Test a point that is not on the line, usually the origin. If it makes the inequality true, shade that side.

Quotient of square roots

The root of a divided by the root of b is the root of a over b.
Combine under one radical, simplify the fraction, then take the root.
If a radical is left in the denominator, rationalize it by multiplying top and bottom by that radical.

Worked

WORKED

y varies directly with x, and y is 12 when x is 3. Write the equation.
Direct means y = kx. Substitute: 12 = k(3), so k = 4.
ANSWER: y = 4x

WORKED

y varies inversely with x, and y is 6 when x is 2. Find y when x is 4.
Inverse means xy = k. So k = 2 × 6 = 12.
Now 4y = 12.
ANSWER: y = 3

Terms

Term	Meaning
direct variation	y = kx, the two quantities rise and fall together
inverse variation	xy = k, one rises as the other falls and the product holds
constant of variation	the k that stays the same across every pair of values

Check yourself

1. y varies directly with x and y is 20 when x is 5. Find k.

2. y varies inversely with x and y is 8 when x is 3. Find k.

3. Solid boundary line or dashed for y is greater than or equal to 2x?

reveal answers

k = 4 · k = 24 · solid, because it includes equals

This page is the summary, not the lesson. The full working, every step of every example, is on the board in class. Come here to remember what the week was about. Go there to learn how it is done.