91Ó°ÊÓ

In the context of an exponential function, a horizontal asymptote is a line that the graph of the function approaches as the independent variable (often x) goes to infinity or negative infinity. This means the function will get closer and closer to a certain value but will never actually reach it. For the given exponential function, we have the equation:

\[f(x) = b a^{-x} + c\]

The horizontal asymptote is represented by the constant term, which is \(c\) in the equation. In our problem, we are given that the horizontal asymptote is \(y=32\). This tells us that as \(x\) becomes very large in the positive or negative direction, \(f(x)\) will approach 32. Thus, we can directly identify that \(c = 32\). This horizontal line at \(y=32\) serves as a boundary the function will closely approach but will not surpass.

The y-intercept of a function is the point where the graph crosses the y-axis. This occurs when the value of \(x\) is zero. It gives us important information about the starting value of the function. For our exponential equation \(f(x) = b a^{-x} + c\), at the y-intercept, we substitute \(x = 0\):

\[f(0) = b a^0 + c\]

Since \(a^0 = 1\), the equation simplifies to \(f(0) = b + c\). Given the y-intercept as 212, we set up the equation:

\[b + 32 = 212\]

Solving this, we find that \(b = 180\). Thus, the y-intercept gives us the sum of \(b\) and the horizontal asymptote \(c\). Solving for \(b\) helps to build the equation further.

Solving for coefficients in an exponential function involves using given points or conditions to calculate unknown parameters in the equation. With initially identified values \(b = 180\) and \(c = 32\), we still need \(a\). Using the point \((2,112)\), where the function value \(f(x)\) should equal 112 for \(x = 2\), we substitute into the function:

\[180 a^{-2} + 32 = 112\]

First, subtract 32 from both sides to solve for \(180 a^{-2}\):

\[180 a^{-2} = 80\]

To isolate \(a^{-2}\), divide both sides by 180:

\[a^{-2} = \frac{4}{9}\]

Next, take the reciprocal of both sides to find \(a^2\):

\[a^2 = \frac{9}{4}\]

Finally, take the square root to solve for \(a\):

\[a = \frac{3}{2}\]

Thus, the process of plugging in known values into the function helps in finding the unknown coefficient \(a\). This step is crucial for developing the complete function.

Identifying a point on the curve involves using a specific coordinate pair that lies on the graph of the exponential function. For the function \(f(x) = b a^{-x} + c\), being given a point \(P(2,112)\) means that when \(x = 2\), the function should output 112. To ensure the point lies correctly on the constructed function curve, we verify by substitution:

\[180 \left(\frac{3}{2}\right)^{-2} + 32 = 112\]

Simplify the power term \(\left(\frac{3}{2}\right)^{-2}\) as follows: Find the reciprocal of \(\left(\frac{3}{2}\right)^2\):

\[(\frac{2}{3})^2 = \frac{4}{9}\]

Multiply by 180:

\[180 \times \frac{4}{9} = 80\]

Then add back the horizontal asymptote:

\[80 + 32 = 112\]

This confirms our calculations are correct, and the point \(P(2,112)\) indeed lies on the graph of the function. Using given points to verify your function is an essential step to ensure the function matches all conditions.