Page 477 - Algebra 1

INVE7STIGATION
Comparing Direct and Inverse Variation
Many situations involve direct or inverse variation. The formula d = r · t, which relates distance, speed, and time, can represent either direct or inverse variation depending on the variable that is held constant.
A direct variation is a relationship between two variables whose ratio is constant. The equation y = kx, where k is a nonzero constant called the constant of variation, shows direct variation between variables x and y.
Identify the constant of variation, given that y varies directly with x. Then write the equation of variation.
1. y is 10 when x is 2. 2. y is 3 when x is 6.
The equation of direct variation can be written equivalently as _y = k.
Distance Alex walks 3 miles per hour. If he walks at that rate for twice as long, he will travel twice as far. The ratio of the distance and time is always the same.
3. Identify the constant of variation.
4. Model Write an equation of direct variation that relates Alex’s time to
his distance traveled.
In inverse variation, when x increases, y decreases. An inverse variation describes a relationship between two variables whose product is a constant. The equation xy = k, where k is a nonzero constant, defines an inverse variation between x and y.
Identify the constant of variation, given that y varies inversely with x. Then write the equation of variation.
5 . y i s 1 w h e n x i s 3 . 6 . y i s 4 w h e n x i s _1 . 2
The equation of inverse variation can be written equivalently as y = _k . x
Notice the difference between the equations for inverse and direct variation. Inverse variation is the quotient of k and x, whereas direct variation is the product of k and x.
Rate Alexlives4milesfromschool.Ifhewalksataslowerratethan normal, it will take him longer to reach his destination. In other words, the more time he spends walking home, the slower he is actually walking. This situation represents inverse variation.
7. Identify the constant of variation.
8. Model Write an equation of inverse variation that relates Alex’s time to his rate of speed.
Math Language
The phrase held constant means that a variable is given a particular value. For example, if r is held constant with a value
of 55, then d = 55t in the expression d = rt.
x
Math Reasoning
Justify Show that the two equations for inverse variation are equivalent.
Online Connection www.SaxonMathResources.com
462 Saxon Algebra 1