Example Card: Identifying an Exponential Function
Identify exponential functions in Grade 9 algebra by checking for a constant ratio (multiplier) between consecutive y-values in a table, confirming the form y=a·bˣ with a constant growth factor.
Key Concepts
Let's see if we can find a consistent growth pattern hiding in this set of points. This example shows how to use the core definition of an exponential function to test a set of points.
Example Problem Determine if the set of ordered pairs satisfies an exponential function: $\{ (0, 5), ( 2, \frac{5}{4}), (1, 10), ( 1, \frac{5}{2}) \}$.
Step by Step 1. First, we arrange the ordered pairs so that the x values are in increasing order. $$ \{ ( 2, \frac{5}{4}), ( 1, \frac{5}{2}), (0, 5), (1, 10) \} $$ 2. We can see that the x values increase by a constant amount of 1. 3. Now, we check if there is a common ratio between the y values. We do this by dividing each y value by the one before it. $$ \frac{5}{2} \div \frac{5}{4} = \frac{5}{2} \times \frac{4}{5} = 2 $$ $$ 5 \div \frac{5}{2} = 5 \times \frac{2}{5} = 2 $$ $$ 10 \div 5 = 2 $$ 4. Because the ratio between consecutive y values is constant, the set of ordered pairs does satisfy an exponential function. The common ratio, or base, is $b=2$.
Common Questions
How do you identify an exponential function from a table of values?
Calculate the ratio of consecutive y-values (divide each y by the previous y). If this ratio is constant throughout the table, the function is exponential. For y-values 2, 6, 18, 54: each ratio is 3, confirming y = 2·3ˣ.
What is the difference between an exponential function and a linear function from a table?
Linear functions have constant differences (add the same amount each step). Exponential functions have constant ratios (multiply by the same factor each step). Check differences first; if not constant, check ratios.
What does the base b represent in an exponential function y = a · bˣ?
The base b is the constant ratio (growth factor). If b > 1, the function models exponential growth. If 0 < b < 1, it models exponential decay. The initial value a is the y-intercept (value when x = 0).