Chapter 5.7, Problem 31PS

Solution

a.

To determine: the function rule that models the ordered pairs of each pattern.

Pattern 1 has a function of $y=x^4$ .

Pattern 2 has a function of $y=4^x$ .

Given information:

Pattern 1: $1,16,81,256,\mathrm{.....}$

Pattern 2: $4,16,64,256,\mathrm{.....}$

Concept Used:

The function rule is the relationship between the input or domain and the output or range. A relation is a function if and only if there exists one value in the range for every domain value.

Calculation:

The first four pairs in pattern 1 are $\left(1,1\right),\left(2,16\right),\left(3,81\right),\left(4,256\right)$ .

Notice that the $y$ -coordinates are equivalent to the $x$ -coordinates multiplied to itself four times.

  $\begin{array}{c}1\cdot 1\cdot 1\cdot 1=1\\ 2\cdot 2\cdot 2\cdot 2=16\\ 3\cdot 3\cdot 3\cdot 3=81\\ 4\cdot 4\cdot 4\cdot 4=256\end{array}$

Therefore, pattern 1 follow $y=x^4$ .

The first four pairs in pattern 2 are $\left(1,4\right),\left(2,16\right),\left(3,64\right),\left(4,256\right)$ .

Notice that the $y$ -coordinates are equivalent to the $x$ -coordinates multiplied to itself four times.

  $\begin{array}{c}{4}^1=4\\ 4^2=16\\ 4^3=64\\ 4^4=256\end{array}$

Therefore, pattern 1 follow $y=4^x$ .

b.

To graph: the function $y=x^4$ and $y=4^x$ .

The graph is shown below:

  

The function $y=4^x$ increases more rapidly than the other function $y=x^4$ .

Given information:

Pattern 1 has a function of $y=x^4$ .

Pattern 2 has a function of $y=4^x$ .

Concept Used:

Polynomial Function: A single term or the sum of two or more terms containing variables with positive whole-number exponents. You cannot have a variable in the denominator (this means that you can't divide by $x$ ).

Exponential Function: This type of function has an $x$ as exponent.

Calculation:

The first four pairs in pattern 1 are $\left(1,1\right),\left(2,16\right),\left(3,81\right),\left(4,256\right)$ .

The first four pairs in pattern 2 are $\left(1,4\right),\left(2,16\right),\left(3,64\right),\left(4,256\right)$ .

Using the ordered pair, plot the graph of pattern 1 and pattern 2 functions.

The graph is shown below:

  

From the above graph, it is observed that the function $y=4^x$ increases more rapidly than the other function $y=x^4$ .

c.

To interpret: the value of $x$ increases will a quantity that is increasing exponentially eventually exceed a quantity that is increasing as a polynomial function.

The pattern 2 which is exponential in nature exceeds pattern 1 which is polynomial in nature.

Given information:

Pattern 1: $1,16,81,256,\mathrm{.....}$

Pattern 2: $4,16,64,256,\mathrm{.....}$

Concept Used:

Polynomial Function: A single term or the sum of two or more terms containing variables with positive whole-number exponents. You cannot have a variable in the denominator (this means that you can't divide by $x$ ).

Exponential Function: This type of function has an $x$ as exponent.

Calculation:

The graph is shown below:

  

From the above graph, it is observed that the pattern 2 is exponential in nature exceeds pattern 1 which is polynomial in nature.