A.CED.2. Writing linear (y = mx + b), quadratic (y = ax2), and exponential (y = a(b)x)
functions
When looking at a table, you can determine if a function is linear, quadratic, or
exponential by looking at the pattern of y-values.
When the x-values in a table are consecutive, you can compare successive y-values to
determine which type of function the table describes.
Linear (y = mx + b)
You can test whether a function is linear by finding the differences between
successive y-values, which are called the first differences.
-If the first differences are all equal, the function is linear.
-If the first differences are not all equal, the function can’t be linear
Example:
x
1
2
3
4
5
y
5
8
11
14
17
1st d
Equation
y = 3x + 2
+3
+3
+3
+3
The 1st difference represents the slope (m). Plug in any value from the table for x and y
and solve for b to find the y-intercept (b).
Quadratic (y = ax2)
You can test whether a function is quadratic by finding the differences between
successive first differences, which are called the
second differences.
-If the second differences are all equal, the function is quadratic.
-If the second differences are not all equal, the function can’t be quadratic
Example:
x
1
2
3
4
5
y
3
12
27
48
75
1st d
+9
+15
+21
+27
2nd d
+6
+6
+6
Equation
y = 3x2
Once you have determined your table represents a quadratic, plug in any value for x and
y from the table and solve for ‘a’ to find the equation.
Exponential (y = abx)
You can test whether a function is exponential by finding the ratios between
successive y-values. If the ratios are all equal, the function is exponential.
If a function is exponential, the base (b) is equal to the common ratio.
Example:
x
1
2
3
4
5
y
9
27
81
243
729
Ratio
*3
*3
*3
*3
Equation
y = 3(3)x
Once you find the ratio and plug it in for ‘b’ in the equation; use any value for x and y from the
table and solve for ‘a’.