1.4 direct variation and proportion objectives: write and apply direct variation equations. write...
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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. *
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.