Answer:
[tex]\text{Average rate of change of }f(x)=x^2+3x-4 \text{ is 8}[/tex]
[tex]\text{Average rate of change of }f(x)=-12(x+2)^2+5 \text{ is -24}[/tex]
Option B: [tex]5.3^x[/tex] grows at the fastest rate for increasing values of x
Step-by-step explanation:
Given the function
[tex]f(x)=x^2+3x-4[/tex]
we have to find the average rate of change for the quadratic function from x=−3 to x = 8
[tex]f(-3)=(-3)^2+3(-3)-4=9-9-4=-4[/tex]
[tex]f(8)=8^2+3(8)-4=64+24-4=84[/tex]
[tex]\text{Average rate of change=}\frac{f(x_2)-f(x_1)}{x_2-x_1}=\frac{84-(-4)}{8-(-3)}=\frac{88}{11}=8[/tex]
Now, given function is
[tex]f(x)=-12(x+2)^2+5 [/tex]
we have to find the average rate of change for the quadratic function from x=−3 to x = 1
[tex]f(-3)=-12(-3+2)^2+5=-12+5=-7[/tex]
[tex]f(1)=-12(1+2)^2+5=-12(9)+5=-108+5=-103[/tex]
[tex]\text{Average rate of change=}\frac{f(x_2)-f(x_1)}{x_2-x_1}=\frac{-103-(-7)}{1-(-3)}=\frac{-96}{4}=-24[/tex]
Now, we have to choose the function which grows at the fastest rate for increasing values of x.
Since we know that quadratic function grows faster than a linear function and an exponential function grows faster than a quadratic function, so the function of an exponential function will have the fastest rate for increasing value of x.
Since an exponential function is in form:
[tex]y=a.b^x[/tex]
From the given choices we can see that function represented by option A is a linear function, function represented by option B is exponential function and function represented by option C is a quadratic function.
Option B: [tex]5.3^x[/tex]
∴ option B is the correct choice.
The average rate of change for the quadratic function from x=−3 to x = 8 is 8.
The function grows at the fastest rate for increasing values of x is [tex]\rm f(x)=5\times 3^x[/tex]
Given
The given function is;
[tex]\rm f(x)=x^2+3x-4[/tex]
The formula to determine the average value of the function is;
[tex]\rm Average \ value = \dfrac{f(x_1)-f(x_2)}{x+y}[/tex]
The value of the function when x = -3 is;
[tex]\rm f(x)=x^2+3x-4\\\\\rm f(-3)=(-3)^2+3(-3)-4\\\\f(-3) = 9-9-4\\\\f(-3)=-4[/tex]
The value of the function when x = 8 is;
[tex]\rm f(x)=x^2+3x-4\\\\\rm f(8)=(8)^2+3(8)-4\\\\f(-3) =64+24-4\\\\f(-3)=84[/tex]
Therefore,
The average rate of change for the quadratic function from x=−3 to x = 8 is;
[tex]\rm Average \ value = \dfrac{f(x_1)-f(x_2)}{x+y}\\\\\rm Average \ value = \dfrac{84-(-4)}{8-(-3)}\\\\\rm Average \ value = \dfrac{88}{11}\\\\\rm Average \ value = 8[/tex]
The exponential function grows the fastest;
Then,
The function is;
[tex]\rm f(x)=5\times 3^x[/tex]
Hence, the function grows at the fastest rate for increasing values of x is [tex]\rm f(x)=5\times 3^x[/tex].
To know more about the Average rate click the link given below.
https://brainly.com/question/18100612
