Question

How do you know if the following data set is exponential: (0,120), (1, 180), (2, 270), (3, 405)?

Answer 1

To check whether the data sets (0,120), (1, 180), (2, 270), (3, 405) are exponential, we have to express the dataset in exponential form. We can see that the x coordinates of the data increases. So we can write the y coordinate in the form $y=120{{k}^{x}}$ .

Complete step-by-step answer:
We have to check whether the data sets (0,120), (1, 180), (2, 270), (3, 405) are exponential. This means that we have to express the dataset in exponential form. Let us consider (0,120). We can see that here, the value of x is 0. So, we have got the value of y when we do an operation involving an exponent such that the power is of x. That is,
$y=120{{k}^{x}}=120{{k}^{0}}=120$ .
Hence, we can write the data (0,120) as $\left( 0,120{{k}^{0}} \right)$
Now, consider the data (1, 180). We have to get $y=180$ when we use the equation $y=120{{k}^{x}}$ . So we have to find the value of k. We know that in this data, $x=1\text{ and }y=180$ . Hence, we can write
$\begin{aligned} & y=120{{k}^{x}} \\\ & \Rightarrow 180=120{{k}^{1}} \\\ \end{aligned}$
Let us solve for k.
$\begin{aligned} & \Rightarrow 180=120k \\\ & \Rightarrow k=\dfrac{180}{120}=\dfrac{18}{12}=\dfrac{3}{2} \\\ \end{aligned}$
Hence, we can write the data (0, 120) as $\left( 0,120{{\left( \dfrac{3}{2} \right)}^{0}} \right)$
Similarly, we will get $\left( 0,120{{\left( \dfrac{3}{2} \right)}^{1}} \right)=\left( 1,180 \right)$
Let us check (2, 270) in the form $\left( 2,120{{k}^{x}} \right)$ .
$\Rightarrow \left( 2,120{{k}^{x}} \right)=\left( 2,120{{\left( \dfrac{3}{2} \right)}^{2}} \right)$
Let us solve this.
$\begin{aligned} & \Rightarrow \left( 2,120\left( \dfrac{9}{4} \right) \right) \\\ & =\left( 2,30\times 9 \right) \\\ & =\left( 2,270 \right) \\\ \end{aligned}$
Hence, we can write $\left( 2,120{{\left( \dfrac{3}{2} \right)}^{2}} \right)=\left( 2,270 \right)$ .
Now, let us consider the data (3, 405) . Let us write this data in the form $\left( 3,120{{k}^{x}} \right)$
$\Rightarrow \left( 3,120{{k}^{x}} \right)=\left( 3,120{{\left( \dfrac{3}{2} \right)}^{3}} \right)$
Let us solve this.
$\begin{aligned} & \Rightarrow \left( 3,120\left( \dfrac{27}{8} \right) \right) \\\ & =\left( 2,30\times \dfrac{27}{2} \right) \\\ & =\left( 2,15\times 27 \right) \\\ & =\left( 2,405 \right) \\\ \end{aligned}$
Hence, we can write $\left( 3,120{{\left( \dfrac{3}{2} \right)}^{3}} \right)=\left( 3,405 \right)$ .
Therefore, we can see that the data sets (0,120), (1, 180), (2, 270), (3, 405) are exponential.

Note: We can call a function as an exponential if we can write it in the form $f\left( x \right)={{a}^{x}}$ , where the value of a must be greater than 1 and a should not be equal to 1.x can be any real number. We have to note that x is a variable while a is a constant.