The general form of an exponential function is given by $y=a\cdot b^x$, where $a$ and $b$ are constants. To find the equation, we need to find the values of $a$ and $b$ that make the function pass through the given points.
Let's start by examining the pattern between the $x$ and $y$ values in the given table.
When $x=0$, $y=3$. We can substitute these values into the equation $y=a\cdot b^x$:
$$3=a\cdot b^0$$
$$3=a\cdot 1$$
$$a=3$$
So, we have determined that $a=3$. The equation so far is $y=3\cdot b^x$.
When $x=2$, $y=0.75$. We can substitute these values into the equation:
$$0.75=3\cdot b^2$$
Next, when $x=3$, $y=0.375$. We can substitute these values into the equation:
$$0.375=3\cdot b^3$$
Now, let's solve the two equations above for $b$.
Solving equation $0.75=3\cdot b^2$:
$$\dfrac{0.75}{3}=b^2$$
$$b^2=\dfrac{0.25}{1}$$
$$b=\pm\sqrt{\dfrac{1}{4}}$$
$$b=\pm\dfrac{1}{2}$$
Solving equation $0.375=3\cdot b^3$:
$$\dfrac{0.375}{3}=b^3$$
$$b^3=\dfrac{0.125}{1}$$
$$b=\sqrt[3]{\dfrac{1}{16}}$$
$$b=\dfrac{1}{2}$$
Considering both solutions, we find that $b=\dfrac{1}{2}$.
Therefore, the equation of the exponential function represented by the table is $y=3\cdot\left(\dfrac{1}{2}\right)^x$.
Find the equation of the exponential function represented by the table below: 0 3 11.5 2 0.75 3 0.375
1 answer
1 answer