Linear, quadratic and exponential functions differ in how the output of the function changes with regards to the input x.
We can use three things to determine the change in output: the first difference, the second difference, and the common ratio.
x | f(x) | \text{first difference} |
---|---|---|
-2 | -4 | |
-1 | -2 | -2-(-4)=2 |
0 | 0 | 0-(-2)=2 |
1 | 2 | 2-0=2 |
2 | 4 | 4-2=2 |
3 | 6 | 6-4=2 |
x | g(x) | \text{first} \\ \text{difference} | \text{second} \\ \text{difference} |
---|---|---|---|
-2 | -4 | ||
-1 | -2 | 2 | |
0 | 2 | 4 | 4-2=2 |
1 | 8 | 6 | 6-4=2 |
2 | 16 | 8 | 8-6=2 |
3 | 26 | 10 | 10-8=2 |
x | f(x) | \text{common ratio} |
---|---|---|
-2 | \dfrac{1}{4} | |
-1 | \dfrac{1}{2} | \dfrac{1}{2}\div \dfrac{1}{4}=2 |
0 | 1 | 1\div \dfrac{1}{2}=2 |
1 | 2 | 2\div 1=2 |
2 | 4 | 4\div 2=2 |
3 | 8 | 8\div 4=2 |
We can also identify a function by the shape of its graph:
Identify if the function f(x) is linear, quadratic or exponential.
x | -2 | -1 | 0 | 1 | 2 | 3 |
---|---|---|---|---|---|---|
f\left( x \right) | 13 | 1 | -3 | 1 | 13 | 33 |
Find the average rate of change of the function f(x) between x = 2 and x = 6.
Would a linear, quadratic or exponential function best model the number of pieces a cake is cut into, if each piece of cake is cut in half every minute?
Assume you start with a whole cake.
Compare the key features of quadratic and exponential functions.