Which statement best describes the growth rates of the functions below?
[tex]y=4 x^2[/tex]
[tex]y=4^x[/tex]
| x | y |
|---|---|
| 0 | 0 |
| 1 | 4 |
| 2 | 16 |
| 3 | 36 |
| 4 | 64 |
| x | y |
|---|---|
| 0 | 1 |
| 1 | 4 |
| 2 | 16 |
| 3 | 64 |
| 4 | 256 |
A. The quadratic function grows faster than the exponential function over the entire interval; [tex]0\ \textless \ x \leq 4[/tex].
B. The exponential function grows faster than the quadratic function over the entire interval; [tex]0\ \textless \ x \leq 4[/tex].
C. The exponential function grows faster than the quadratic function over only one interval; [tex]2\ \textless \ x \leq 3[/tex].
D. The exponential function grows faster than the quadratic function over two intervals; [tex]2\ \textless \ x \leq 4[/tex].
Answer (1)
Calculate the values of the quadratic function y = 4 x 2 and the exponential function y = 4 x for x = 1 , 2 , 3 , 4 .
Compare the values at each x to determine which function grows faster.
Observe that the exponential function grows faster than the quadratic function for 2 < x ≤ 3 and 3 < x ≤ 4 .
Conclude that the exponential function grows faster over two intervals; 2 < x ≤ 4 .
Explanation
Understanding the Problem We are given two functions, y = 4 x 2 (quadratic) and y = 4 x (exponential), and asked to compare their growth rates over the interval 0 < x ≤ 4 . We have tables of values for both functions at integer values of x from 0 to 4.
Comparing Function Values Let's compare the values of the two functions at each integer value of x in the interval 0 < x ≤ 4 :
For x = 1 :
4 x 2 = 4 ( 1 ) 2 = 4 4 x = 4 1 = 4
For x = 2 :
4 x 2 = 4 ( 2 ) 2 = 16 4 x = 4 2 = 16
For x = 3 :
4 x 2 = 4 ( 3 ) 2 = 36 4 x = 4 3 = 64
For x = 4 :
4 x 2 = 4 ( 4 ) 2 = 64 $4^x = 4^4 = 256
Analyzing Growth Rates Now, let's analyze the growth rates based on these values:
From x = 1 to x = 2 , both functions increase from 4 to 16, so their growth is the same.
From x = 2 to x = 3 , the quadratic function increases from 16 to 36, while the exponential function increases from 16 to 64. The exponential function grows faster in this interval ( 2 < x ≤ 3 ).
From x = 3 to x = 4 , the quadratic function increases from 36 to 64, while the exponential function increases from 64 to 256. The exponential function grows faster in this interval ( 3 < x ≤ 4 ).
Therefore, the exponential function grows faster than the quadratic function over two intervals: 2 < x ≤ 3 and 3 < x ≤ 4 . This can be summarized as 2 < x ≤ 4 .
Final Answer The exponential function grows faster than the quadratic function over two intervals; $2
