compare the magnitude of the estimated average rates of change of the exponential function pictured above…

Answer:

1. Average rate of change of the quadratic function (f(x)=x^{2}-20):
   • The formula for the average rate of change of a function (y = f(x)) over the interval ([a,b]) is (\frac{f(b)-f(a)}{b - a}).
   • Here, (a=-9), (b = 2), (f(x)=x^{2}-20).
   • First, find (f(-9)) and (f(2)):
     • (f(-9)=(-9)^{2}-20=81 - 20=61).
     • (f(2)=2^{2}-20=4 - 20=-16).
   • Then, calculate the average - rate of change:
     • (\frac{f(2)-f(-9)}{2-(-9)}=\frac{-16 - 61}{2 + 9}=\frac{-77}{11}=-7). The magnitude is (| - 7|=7).
2. Estimate the average rate of change of the exponential function:
   • The exponential function passes through the points ((-9,-1)) and ((2,3)) (estimated from the graph).
   • Using the average - rate of change formula (\frac{y_{2}-y_{1}}{x_{2}-x_{1}}), where ((x_{1},y_{1})=(-9,-1)) and ((x_{2},y_{2})=(2,3)).
   • (\frac{3-(-1)}{2-(-9)}=\frac{3 + 1}{2 + 9}=\frac{4}{11}\approx0.36). The magnitude is (|\frac{4}{11}|\approx0.36).
   • Since (7>0.36), the quadratic function (f(x)=x^{2}-20) has a greater magnitude of the average rate of change over the interval ([-9,2]).

Explanation:

Step1: Calculate quadratic function value at endpoints

(f(-9)=(-9)^{2}-20 = 61), (f(2)=2^{2}-20=-16)

Step2: Compute quadratic function average rate of change

(\frac{f(2)-f(-9)}{2-(-9)}=\frac{-16 - 61}{11}=-7), magnitude (| - 7| = 7)

Step3: Identify exponential function points

Points are ((-9,-1)) and ((2,3))

Step4: Calculate exponential function average rate of change

(\frac{3-(-1)}{2-(-9)}=\frac{4}{11}\approx0.36), magnitude (|\frac{4}{11}|\approx0.36)

Step5: Compare magnitudes

Since (7>0.36), quadratic function has greater rate of change.