A function, f, can be written in the form f(x) = ax + b, where a and b are numbers.

Question

The table below shows the values of f (x) for different values of x.
Use this information to work out the values of a and b.
Give each of your answers as an integer or a decimal.
x = 0, 1, 2, 3
f(x) = 3, 7, 11, 15

Answer 1

To solve for a and b, we need to first determine the rate of change in the function f(x). This rate of change corresponds to the value of a in the function f(x) = ax + b.

Given the values of f(x) for x = 0, 1, 2, 3
f(0) = 3
f(1) = 7
f(2) = 11
f(3) = 15

To find the rate of change, we calculate the difference in f(x) values for subsequent x values:
f(1) - f(0) = 7 - 3 = 4
f(2) - f(1) = 11 - 7 = 4
f(3) - f(2) = 15 - 11 = 4

Since the rate of change is consistent for every increment of x (4), the value of a in the function f(x) = ax + b is 4.

Now we can substitute the value of a back into the function using one of the points provided. Let's use x = 0, f(x) = 3:
3 = 4 * 0 + b
b = 3

Therefore, the values of a and b in the function f(x) = ax + b are:
a = 4
b = 3