Question

use the table to answer the question. which exponential equation contains the points shown in the input - output table? table of values for a exponential equation

x	y
1	0.625

( point)
$\bigcirc$ $y = 0.0625^{x}$
$\bigcirc$ $y = 125\cdot - 0.005^{x}$
$\bigcirc$ $y = 125\cdot 0.005^{x}$
$\bigcirc$ $y = - 125\cdot 0.005^{x}$
graphing calculator

Explanation:

Step1: Recall the general form of an exponential equation

The general form of an exponential equation is \( y = ab^x \), where \( a \) is the initial value and \( b \) is the base.

Step2: Substitute \( x = 1 \) and \( y = 0.625 \) into the general form

For \( x = 1 \), \( y = ab^1=ab \). So \( ab = 0.625 \).

Step3: Substitute \( x=- 2\) and \( y = 5000000\) into the general form

For \( x=-2\), \( y=ab^{-2}=\frac{a}{b^{2}}\). So \(\frac{a}{b^{2}}=5000000\).

Step4: Solve the system of equations

From \( ab = 0.625\), we get \( a=\frac{0.625}{b}\). Substitute this into \(\frac{a}{b^{2}}=5000000\):

\[

$$\begin{align*} \frac{\frac{0.625}{b}}{b^{2}}&=5000000\\ \frac{0.625}{b^{3}}&=5000000\\ b^{3}&=\frac{0.625}{5000000}\\ b^{3}&=1.25\times 10^{-7}\\ b^{3}&=\frac{1.25}{10^{7}}\\ b^{3}&=\frac{1}{8\times 10^{6}}\\ b&=\frac{1}{200}\\ b& = 0.005 \end{align*}$$

\]

Then substitute \( b = 0.005\) into \( ab=0.625\):

\(a\times0.005 = 0.625\), so \(a=\frac{0.625}{0.005}=125\)

So the equation is \( y = 125\times0.005^{x}\)

We can also check the sign. When \( x=-2\), \(y = 125\times(0.005)^{-2}=125\times\frac{1}{(0.005)^{2}}=125\times\frac{1}{2.5\times 10^{-5}}=125\times40000 = 5000000\), which matches. When \( x = 1\), \(y=125\times0.005=0.625\), which matches. Also, the base of an exponential function \( b>0\) and \(b
eq1\), so we can eliminate the option with negative base or negative coefficient (since if the coefficient is negative, when \( x = - 2\), the value will be negative, but our \( y = 5000000>0\)). So the correct option is \( y=125\cdot0.005^{x}\)

Answer:

C. \( y = 125\cdot0.005^{x}\) (Here we assume the options are labeled as A: \(y = 0.0625^{x}\), B: \(y=125\cdot - 0.005^{x}\), C: \(y = 125\cdot0.005^{x}\), D: \(y=-125\cdot0.005^{x}\))