Answer:
1) The average rate of change is 8.
2) The average rate of change is 112.
3) f(x) grows at the fastest rate.
Step-by-step explanation:
1) Given : Function \(f(x)=x^2+3x-4\)
To find : What is the average rate of change for the quadratic function from x=−3 to x = 8?
Solution :
The average rate of change is \(A_v=\frac{f(b)-f(a)}{b-a}\)
Where, a=-3 and b=8
\(A_v=\frac{((8)^2+3(8)-4)-((-3)^2+3(-3)-4)}{8-(-3)}\)
\(A_v=\frac{(84)-(-4)}{11}\)
\(A_v=\frac{88}{11}\)
\(A_v=8\)
The average rate of change is 8.
2) Given : Function \(f(x)=-14(x+4)^2-8\)
To find : What is the average rate of change for the quadratic function from x=−2 to x = 2?
Solution :
The average rate of change is \(A_v=\frac{f(b)-f(a)}{b-a}\)
Where, a=-2 and b=2
\(A_v=\frac{(-14(2+4)^2-8)-(-14(-2+4)^2-8)}{-2-2}\)
\(A_v=\frac{(-512)-(-64)}{-4}\)
\(A_v=\frac{-448}{-4}\)
\(A_v=112\)
The average rate of change is 112.
3) To find : Which function grows at the fastest rate for increasing values of x?
Solution :
1) \(f(x)=4\cdot2^x\) is an exponential function.
2) \(h(x)=9x^2+25\) is a quadratic function.
3) \(g(x)=15x+6\) is a linear function.
Increasing values of x will increase exponentially.
So, it will grow at the fastest rate.
f(x) grows at the fastest rate.
