Use the graph of $y=2^{x}$ and transformations to sketch the exponential function $f(x)=2^{x}-3$. Determine the domain and range. Also, determine the $y$-intercept, and find the equation of the horizontal asymptote. Use the coordinates of the three points of the graph of $y=2^{x}$ to determine the corresponding points that lie on the graph of $f(x)=2^{x}-3$. \begin{tabular}{|l|l|l|l|} \hline Points that lie on the graph of $y=2^{x}$ & $\left(-1, \frac{1}{2}\right)$ & $(0,1)$ & $(1,2)$ \\ \hline Corresponding points that lie on the graph of $f(x)=2^{x}-3$ & $\square$ & $\square$ & $\square$ \\ \hline \end{tabular} (Type ordered pairs, using integers or fractions. Simplify your answers.)

Step 1 ：The function \(f(x)=2^{x}-3\) is a transformation of the function \(y=2^{x}\). Specifically, it is a vertical shift of the graph of \(y=2^{x}\) downwards by 3 units. This means that the y-coordinate of each point on the graph of \(y=2^{x}\) will be decreased by 3 to get the corresponding point on the graph of \(f(x)=2^{x}-3\).

Step 2 ：To find the corresponding points that lie on the graph of \(f(x)=2^{x}-3\), we can subtract 3 from the y-coordinate of each point on the graph of \(y=2^{x}\).

Step 3 ：The domain of the function \(f(x)=2^{x}-3\) is the same as the domain of the function \(y=2^{x}\), which is all real numbers. The range of the function \(f(x)=2^{x}-3\) is all real numbers less than -3, because the graph of \(y=2^{x}\) is always above the x-axis, and shifting it downwards by 3 units will make it always below the line \(y=-3\).

Step 4 ：The y-intercept of the function \(f(x)=2^{x}-3\) is the y-coordinate of the point where the graph intersects the y-axis. This occurs when \(x=0\), so we can substitute \(x=0\) into the function to find the y-intercept.

Step 5 ：The horizontal asymptote of the function \(f(x)=2^{x}-3\) is the line \(y=-3\), because the graph approaches this line as \(x\) goes to negative infinity.

Step 6 ：The corresponding points that lie on the graph of \(f(x)=2^{x}-3\) are \(\boxed{(-1, -2.5)}\), \(\boxed{(0, -2)}\), and \(\boxed{(1, -1)}\). The y-intercept of the function is \(\boxed{-2}\).

Step 7 ：The domain of the function is all real numbers, and the range is all real numbers less than -3. The equation of the horizontal asymptote is \(y=\boxed{-3}\).