Question

Describe the transformation of \(f\) represented by \(g\). Then graph each function. \(f(x)=e^{-x}, g(x)=e^{-x}-9\)

Short answer

The function \(g(x) = e^{-x} - 9\) is a vertical translation of the function \(f(x) = e^{-x}\) downwards by 9 units. When graphed, the most notable difference between the two is a shift down along the y-axis for the graph of \(g(x)\) compared to \(f(x)\).

Step-by-step solution

Understand the Transformation

The functions \(f(x)=e^{-x}\) and \(g(x)=e^{-x}-9\) are to compared. Notice that the function \(g(x)\) is simply obtained from \(f(x)\) by subtracting a constant 9. This translates to shifting function \(f(x)\) downward by 9 units. So, \(g(x)=e^{-x}-9\) is just \(f(x)=e^{-x}\) shifted 9 units down.

Plot the Original Function

Now, the function \(f(x)=e^{-x}\) is to be plotted. The general shape of the function \(e^{-x}\) is a decreasing exponential curve. The curve starts at a high value when \(x\) is very negative, and decreases as \(x\) gets larger, approaching but never reaching the \(x\)-axis, which it asymptotically touches. The graph passes through the point (0,1), since \(e^{0} = 1\).

Plot the Transformed Function

Next, function \(g(x) = e^{-x}-9\) needs to be plotted. As established above, this graph will have the same shape as that of \(f(x)\), but will simply be shifted 9 units down. So, instead of passing through the point (0,1), it will pass through the point (0,-8), as \(e^{0} - 9 = -8\).

Compare the Two Graphs

Comparing the two graphs together, it is obvious that the graph of \(g(x)\) is indeed just the graph of \(f(x)\) shifted vertically downwards by 9 units.

Key concepts

Exponential Decay

The concept of exponential decay plays a critical role in understanding functions like f(x)=e^{-x}. It describes how a quantity decreases rapidly at first, but then more slowly over time. Mathematically, the base of the exponential function is between 0 and 1, which results in a graph that approaches the x-axis asymptotically but never actually reaches zero.

Intuitively, think of it like a car decelerating: initially, the drop in speed is noticeable, but as the car moves slower, the decrease in speed feels less drastic. The function f(x)=e^{-x} is a classic example of exponential decay, where the number e (approximately 2.718) is raised to the negative x, resulting in a decline as x increases. This decay is fundamental to many natural processes, from radioactive decay in physics to depreciation in finance.

Vertical Shift

The vertical shift of an exponential function is a transformation that moves the graph up or down along the y-axis without changing its shape. This is exactly what happens with g(x)=e^{-x}-9, reflecting a downward shift of 9 units from the original function f(x)=e^{-x}.

Imagine picking up a graph and simply moving it straight up or down. Just like moving a painting on a wall, the image remains the same, just in a different position. When the vertical shift is downward, as with subtracting 9, every point on the graph goes down by that amount. If y was 0 before the shift, it will be -9 after, as observable in the point moving from (0,1) to (0,-8).

Graphing Exponential Functions

The process of graphing exponential functions requires a clear understanding of its components. For the function f(x)=e^{-x}, you start by marking known points, such as (0,1), since e to the power of zero is 1. You also need to consider the end behavior: as x goes to negative infinity, f(x) grows without bound, and as x goes to positive infinity, f(x) approaches zero.

Next, taking into account transformations like the vertical shift in g(x)=e^{-x}-9, you adjust these points and the end behavior accordingly. Tools such as a graphing calculator or software can be hugely beneficial in visualizing these functions, especially when dealing with transformations.

Asymptotic Behavior

The term asymptotic behavior describes how a graph behaves as it gets closer to a line called an asymptote, which it never touches. For exponential decay functions like f(x)=e^{-x}, the x-axis (or y = 0) is a horizontal asymptote. The function tapers off as it approaches this line, but no matter how far along the x-axis you travel, the graph never actually reaches the asymptote.

Understanding this concept is crucial when graphing because it tells us that the values of f(x) become infinitesimally close to zero as x increases. It's a concept that not only applies to exponential functions but also to various types of algebraic and transcendental functions, reflecting the idea that some factors can get close to a value but never actually get there, like a thirsty cat trying to reach a water stream running just out of reach.