find the rate of change of
f(x)-x^2+4
a. from -2 to 0
b. from 1 to 3
c. from -2 to 1
please show work
To find the rate of change of a function, we need to calculate the slope of the function between two given points. The slope of a function represents how much the function is changing over a given interval.
Given the function f(x) = x^2 + 4, we will calculate the rate of change for different intervals as follows:
a. From -2 to 0:
To find the rate of change between -2 and 0, we need to calculate the difference in the function values and divide it by the difference in x-values.
First, let's find the value of f(x) at -2:
f(-2) = (-2)^2 + 4 = 4 + 4 = 8
Next, let's find the value of f(x) at 0:
f(0) = (0)^2 + 4 = 0 + 4 = 4
Now, let's calculate the rate of change:
Rate of Change = (f(0) - f(-2)) / (0 - (-2)) = (4 - 8) / (0 + 2) = -4 / 2 = -2
Therefore, the rate of change of f(x) from -2 to 0 is -2.
b. From 1 to 3:
Similarly, let's find the values of f(x) at 1 and 3:
f(1) = (1)^2 + 4 = 1 + 4 = 5
f(3) = (3)^2 + 4 = 9 + 4 = 13
Now, let's calculate the rate of change:
Rate of Change = (f(3) - f(1)) / (3 - 1) = (13 - 5) / (3 - 1) = 8 / 2 = 4
Therefore, the rate of change of f(x) from 1 to 3 is 4.
c. From -2 to 1:
Similarly, let's find the values of f(x) at -2 and 1:
f(-2) = (-2)^2 + 4 = 4 + 4 = 8
f(1) = (1)^2 + 4 = 1 + 4 = 5
Now, let's calculate the rate of change:
Rate of Change = (f(1) - f(-2)) / (1 - (-2)) = (5 - 8) / (1 + 2) = -3 / 3 = -1
Therefore, the rate of change of f(x) from -2 to 1 is -1.
In summary, the rate of change for the given intervals is:
a. From -2 to 0: -2
b. From 1 to 3: 4
c. From -2 to 1: -1