Arrange these functions from the greatest to the least value based on the average rate of change in the specified interval.
Tiles
f(x) = x2 + 3x
interval: [-2, 3]f(x) = 3x - 8
interval: [4, 5]f(x) = x2 - 2x
interval: [-3, 4]
f(x) = x2 - 5
interval: [-1, 1]
Sequence
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The average rate of change for some function f(x) on an invterval [a, b], x=a to x=b,
can be thought of as the slope value for a line through those two interval endpoints (a, f(a)  and (b, f(b)
slope = rise/run = [ f(b) - f(a)] / [ b - a]
For example the first function over the interval [-2, 3]  -
f(x) = x^2 + 3x
f(-2) = (-2)^2 + 3*-2  = -2
f(3) = 3^2 + 3*3 = 18
so the average rate of change for the function over that interval is
$$\frac{f(3) - f(-2)}{3 - (-2)} = \frac{18 - (-2)}{3 - (-2)} = \frac{20}{5}=4$$
The first one has a value of 4.
Do that same thing for all and order them

we know that

The Average Rate of Change is the slope of a line or a curve on a given range. It is defined as the ratio of the difference in the function f(x) as it changes from ’a’ to ’b’ to the difference between ’a’ and ’b’.

So

$$A=\frac{(f(b)-f(a)}{(b-a)}$$

case a) $$f(x) = x^{2} + 3x$$

in the interval $$[-2, 3]$$

$$a=-2\\ b=3\\ f(a)=(-2)^{2} +3*(-2)\\ f(a)=-2\\ f(b)=(3)^{2} +3*(3)\\ f(b)=18$$

Find the Average Rate of Change

$$A=\frac{(18-(-2))}{(3+2)}$$

$$A=\frac{20}{5}$$

$$A=4$$

case b) $$f(x) = 3x-8$$

in the interval $$[4, 5]$$

$$a=4\\ b=5\\ f(a)=3*4-8\\ f(a)=4\\ f(b)=3*5-8\\ f(b)=7$$

Find the Average Rate of Change

$$A=\frac{(7-(4))}{(5-4)}$$

$$A=\frac{3}{1}$$

$$A=3$$

case c) $$f(x) = x^{2} -2x$$

in the interval $$[-3, 4]$$

$$a=-3\\ b=4\\ f(a)=(-3)^{2} -2*(-3)\\ f(a)=15\\ f(b)=(4)^{2} -2*(4)\\ f(b)=8$$

Find the Average Rate of Change

$$A=\frac{(8-(15))}{(4+3)}$$

$$A=\frac{-7}{7}$$

$$A=-1$$

case d) $$f(x) = x^{2} -5$$

in the interval $$[-1, 1]$$

$$a=-1\\ b=1\\ f(a)=(-1)^{2} -5\\ f(a)=-4\\ f(b)=(1)^{2} -5\\ f(b)=-4$$

Find the Average Rate of Change

$$A=\frac{(-4-(-4))}{(1+1)}$$

$$A=\frac{0}{2}$$

$$A=0$$

Arrange these functions from the greatest to the least value based on the average rate of change in the specified interval

so

1) $$f(x) = x^{2} + 3x$$ -> $$A=4$$
2) $$f(x) = 3x-8$$ -> $$A=3$$
3) $$f(x) = x^{2} -5$$ -> $$A=0$$
4) $$f(x) = x^{2} -2x$$ -> $$A=-1$$