Constructing and Comparing Linear and Exponential Models
Michael bought two doughnut making machines for his factory. The first doughnut machine make doughnuts at a rate of 3 per minute. The second doughnut machine triples that amount of doughnuts every minute. They both start of with three doughnuts.
Machine 1: 3, 6, 9, 12, 15, 18, 21
Machine 2: 3, 9, 27, 81, 243, 729, 2187
Both machines move at constant rates. The first machine moves at a constant rate of positive 2. The second machine moves at a constant percent per unit of positive 2. Knowing this, we know that the first machine shows a linear function, and that the second machine shows an exponential function.
REMEMBER! Constant per percent per unit is for describing a exponential function.
Constant rate is for describing a linear function.
Machine 1: 3, 6, 9, 12, 15, 18, 21
Machine 2: 3, 9, 27, 81, 243, 729, 2187
Both machines move at constant rates. The first machine moves at a constant rate of positive 2. The second machine moves at a constant percent per unit of positive 2. Knowing this, we know that the first machine shows a linear function, and that the second machine shows an exponential function.
REMEMBER! Constant per percent per unit is for describing a exponential function.
Constant rate is for describing a linear function.
y=3x
y=3^x
In the linear model above, the domain is all real whole numbers greater than 0. Since you can't make negative amounts of doughnuts, the y-intercept begins at 0. Also, because you can't make a negative amount of doughnuts both graphs will increase. Both graphs intersect at (1,3) because that is the point at which both machines make the same amount of doughnuts. The y-intercept for the exponential function is 1 because exponential functions can't have a y-intercept lower than one.
In the linear model above, the domain is all real whole numbers greater than 0. Since you can't make negative amounts of doughnuts, the y-intercept begins at 0. Also, because you can't make a negative amount of doughnuts both graphs will increase. Both graphs intersect at (1,3) because that is the point at which both machines make the same amount of doughnuts. The y-intercept for the exponential function is 1 because exponential functions can't have a y-intercept lower than one.