Question

Let f(x)=x2+3x−4 .

Answer 1

Here are a ton of correct answers. I got a few questions wrong but they still show the correct answer.

Answer 2

Answer:

$\text{Average rate of change of }f(x)=x^2+3x-4 \text{ is 8}$

$\text{Average rate of change of }f(x)=-12(x+2)^2+5 \text{ is -24}$

Option B: $5.3^x$ grows at the fastest rate for increasing values of x

Step-by-step explanation:

Given the function

$f(x)=x^2+3x-4$

we have to find the average rate of change for the quadratic function from x=−3 to x = 8

$f(-3)=(-3)^2+3(-3)-4=9-9-4=-4$

$f(8)=8^2+3(8)-4=64+24-4=84$

$\text{Average rate of change=}\frac{f(x_2)-f(x_1)}{x_2-x_1}=\frac{84-(-4)}{8-(-3)}=\frac{88}{11}=8$

Now, given function is

$f(x)=-12(x+2)^2+5$

we have to find the average rate of change for the quadratic function from x=−3 to x = 1

$f(-3)=-12(-3+2)^2+5=-12+5=-7$

$f(1)=-12(1+2)^2+5=-12(9)+5=-108+5=-103$

$\text{Average rate of change=}\frac{f(x_2)-f(x_1)}{x_2-x_1}=\frac{-103-(-7)}{1-(-3)}=\frac{-96}{4}=-24$

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:

$y=a.b^x$

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: $5.3^x$

∴ option B is the correct choice.