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

the answer is

1) \( f(x) = x^{2} + 3x \) -> \( A=4 \)

2) \( f(x) = 3x-8 \) -> \( A=3 \)

3) \( f(x) = x^{2} -5 \) -> \( A=0 \)

4) \( f(x) = x^{2} -2x \) -> \( A=-1 \)


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