Question

given the graph of the exponential equation representing the number of bacteria cells based on time, which option is the correct exponential equation for the graph? (1 point)
$y = 5(\frac{3}{5})^x$
$y=\frac{1}{5}(5)^x$
$y = 3(\frac{1}{3})^x$
$y = 5(\frac{1}{5})^x$

Explanation:

Step1: Recall exponential - function form

The general form of an exponential function is $y = ab^{x}$, where $a$ is the $y$ - intercept and $b$ is the base.

Step2: Determine the y - intercept

The graph passes through the point $(0,5)$. Substituting $x = 0$ into $y=ab^{x}$, we get $y=a\cdot b^{0}=a$. Since $b^{0}=1$ for $b
eq0$, when $x = 0$ and $y = 5$, then $a = 5$.

Step3: Use another point to find the base

The graph also passes through the point $(1,3)$. Substitute $x = 1$, $y = 3$, and $a = 5$ into $y = ab^{x}$. We have $3=5\cdot b^{1}$, so $b=\frac{3}{5}$. The exponential function is $y = 5(\frac{3}{5})^{x}$. But if we assume the general form $y=a(\frac{1}{k})^{x}=a\cdot k^{-x}$, when $x = 0,y = 5$ (so $a = 5$) and when $x = 1,y = 3$, substituting into $y=a\cdot k^{-x}$ gives $3 = 5\cdot k^{-1}$, then $k=\frac{5}{3}$. The function can be written as $y = 5(\frac{3}{5})^{x}=5(\frac{1}{\frac{5}{3}})^{x}$. If we rewrite it in the form of the given options, we know that for an exponential function $y = ab^{x}$, when $x = 0$, $y=a$. From the point $(0,5)$, $a = 5$. And from the point $(1,3)$, when we substitute into $y = ab^{x}$: $3=5b$, $b=\frac{3}{5}$. The general form of an exponential decay function is $y=a(1 - r)^{x}$, where $a$ is the initial value and $1 - r$ is the decay factor. Here $a = 5$ and the decay factor is $\frac{3}{5}$. The correct form is $y = 5(\frac{3}{5})^{x}$. If we rewrite it in terms of the given options, we know that the general form of an exponential function $y = ab^{x}$. Using the point $(0,5)$ we get $a = 5$, and using the point $(1,3)$ we find $b=\frac{3}{5}$. The exponential function is $y = 5(\frac{3}{5})^{x}$.

Answer:

The correct exponential equation for the graph is $y = 5(\frac{3}{5})^{x}$ (but this option is not shown in the provided choices. There might be a mis - typing in the problem setup or options. If we assume the general form $y = ab^{x}$ and use the points $(0,5)$ and $(1,3)$ to solve for $a$ and $b$, we find $a = 5$ and $b=\frac{3}{5}$). If we consider the closest form among the given options based on the process of finding the exponential - function parameters from the points on the graph, we note that when $x = 0$, the value of $y$ gives the coefficient of the exponential term. Since the $y$ - intercept is 5, we can eliminate options where the coefficient is not 5. Without a correct $y = 5(\frac{3}{5})^{x}$ option, there is an issue with the problem as presented. If we assume some error in transcription and we go by the principle of using the $y$ - intercept $(0,5)$ to start with, we know the function should be of the form $y = 5(\text{some base})^{x}$.