Question

quadratic functions
tables of values for two different functions are given.
which statement is true?
the second function grows approximately 6 times faster than the first function over the interval (2 < x < 3).
the second function grows approximately 2.5 times slower than the first function over the interval (2 < x < 3).
the second function grows approximately 21 times faster than the first function over the interval (4 < x < 5).
the second function grows approximately 21 times slower than the first function over the interval (4 < x < 5).

Explanation:

Step1: Analyze the first function (left table)

Looking at the left table, when \(x = 1\), \(y = 4=(2\times1)^2\); \(x = 2\), \(y = 16=(2\times2)^2\); \(x = 3\), \(y = 36=(2\times3)^2\); so the first function is \(y=(2x)^2 = 4x^2\).

Step2: Analyze the second function (right table)

Looking at the right table, when \(x = 1\), \(y = 4 = 4^1\); \(x = 2\), \(y = 16=4^2\); \(x = 3\), \(y = 64 = 4^3\); so the second function is \(y = 4^x\).

Step3: Check the interval \(2

For the first function, at \(x = 2\), \(y_1=4\times(2)^2=16\); at \(x = 3\), \(y_1'=4\times(3)^2 = 36\). The change in \(y\) for the first function over \(2

For the second function, at \(x = 2\), \(y_2 = 4^2=16\); at \(x = 3\), \(y_2'=4^3 = 64\). The change in \(y\) for the second function over \(2

Now, find the ratio \(\frac{\Delta y_2}{\Delta y_1}=\frac{48}{20}=2.4\approx2.5\)? Wait, no, wait. Wait, maybe I made a mistake. Wait, the first function: when \(x = 2\), \(y = 16\); \(x = 3\), \(y = 36\), so \(\Delta y_1=36 - 16 = 20\). The second function: \(x = 2\), \(y = 16\); \(x = 3\), \(y = 64\), \(\Delta y_2=64 - 16=48\). Then \(\frac{\Delta y_2}{\Delta y_1}=\frac{48}{20} = 2.4\approx2.5\)? But the first option says "6 times faster", second says "2.5 times slower" which is wrong, third and fourth: check interval \(4

Step4: Check the interval \(4

For the first function, at \(x = 4\), \(y_1=4\times(4)^2=64\); at \(x = 5\), \(y_1'=4\times(5)^2 = 100\). \(\Delta y_1=100 - 64 = 36\).

For the second function, at \(x = 4\), \(y_2=4^4 = 256\); at \(x = 5\), \(y_2'=4^5=1024\). \(\Delta y_2=1024 - 256=768\).

Now, find the ratio \(\frac{\Delta y_2}{\Delta y_1}=\frac{768}{36}\approx21.33\approx21\). So the ratio of the change in the second function to the first function over \(4

Wait, let's re - check the interval \(2

First function: \(x = 4\), \(y = 64\); \(x = 5\), \(y = 100\), \(\Delta y_1=100 - 64 = 36\).

Second function: \(x = 4\), \(y = 256\); \(x = 5\), \(y = 1024\), \(\Delta y_2=1024 - 256 = 768\).

Ratio \(\frac{768}{36}=\frac{64}{3}\approx21.33\approx21\). So the third statement: "The second function grows approximately 21 times faster than the first function over the interval \(4 < x < 5\)." is correct.

Answer:

The second function grows approximately 21 times faster than the first function over the interval \(4 < x < 5\). (The option corresponding to this statement)