1.4 Direct Variation and Proportion

Objectives: Write and apply direct variation equations. Write and solve proportions.

Standard: 2.8.11.P Analyze a relation to determine whether a direct or inverse variation exists and represent it algebraically and graphically.

I. Determine whether each equation describes a direct variation.

Direct Variation - The variable y varies directly as x.

y = kx is known as a direct-variation equation. k is called the constant of variation.

y = 2x yesy = ½ x yes

y = 2x + 1 no

y = 3/x no

II. y varies directly as x. Find the constant of variation, k, and write an equation of direct variation that relates the two variables, y = kx.

Ex 1. y = -24 when x = 4

Ex 2. y = -16 when x= 2

Ex 3. y = 1 when x = ½

III. y varies directly as x.

Ex. 1 If y is 2.8 when x is 7, find y when x is -4.

Ex. 2 If y is 6.3 when x is 70, find y when x is 5.4. *

III. y varies directly as x.

Ex. 3 If y is -5 when x is 2.5, find y when x is 6. *

IV. Use a direct variation equation to solve each word problem.

Ex 1. If 6 tickets cost $72, find the cost of 10 tickets. *72 = k (6)12 = k

y = 12 (10) y = 120

Ex 2. If 3 CDs on sale cost $18, find the cost of 12 CDs. *

Ex 3. If 8 sodas cost $3.20, find the cost of 20 sodas. *

IV. Use a direct variation equation to solve each word problem.

Ex 4. Each day Jon rides his bicycle for exercise. When traveling a constant rate, he rides 4 miles in about 20 minutes. At this rate, how long would it take Jon to travel 7 miles? Recall that distance, d, rate, r, and elapsed time, t, are related by the equation d = rt. *

Rate = 4 miles/20 minutes = 1/5 miles per minute

d = 1/5 t7 = 1/5 t

35 = t

V. Proportions

If y varies directly with x, then y is proportional to x. A proportion is a statement that two ratios are equal. A ratio is the comparison of 2 quantities by division.

A proportion of the form a = c can be rearranged as follows: b d a = c b d a bd = c bd b d ad = bc

V. Proportions

* Cross-Product Property of Property of Proportion For b 0 and d 0: If a = c, then ad = bc.

b d* In a proportion of the form a/b = c/d; a and d are

the extremes and b and c are the means.

* By the Cross-Product Property, the product of the extremes equals the product of the means.

V. Proportions

Ex 1. w = 10 4 12

Ex 2. 3 = x *

5 2

V. Proportions

Ex 3. 3x –1 = x * 5 2

Ex. 4 Proportions

Homework

Integrated Algebra II- Section 1.4 Level A even #’s

Academic Algebra II- Section 1.4 Level B