Question

Write a rule for \(g\) that represents the indicated transformations of the graph of \(f\). \(f(x)=e^{-x}\); translation 4 units right and 1 unit down, followed by a vertical shrink by a factor of \(\frac{1}{3}\)

Short answer

The rule for \(g\), after applying the specified transformations, is \(g(x) = \frac{1}{3}(e^{-x+4} - 1)\).

Step-by-step solution

Apply the horizontal shift

A shift to the right by 4 units corresponds to replacing \(x\) with \(x-4\) in the function equation. This gives us a new function: \(g_1(x) = e^{-(x-4)} = e^{-x+4}\).

Apply the vertical shift

A shift downwards by 1 unit corresponds to subtracting 1 from the function equation. After applying this transformation to the function we got in Step 1, we get a new function: \(g_2(x) = e^{-x+4} - 1\).

Apply the vertical shrink

A vertical shrink by a factor of \(\frac{1}{3}\) corresponds to multiplying the function equation by \(\frac{1}{3}\). Applying this to the function from Step 2 gives us the final function: \(g(x) = \frac{1}{3}(e^{-x+4} - 1)\).

Key concepts

Horizontal Shift in Function

A horizontal shift occurs in a function when every point on the graph is moved to the left or right, resulting in a lateral movement of the entire graph. This is one of the most common transformations students will encounter.

To understand it with a formula, consider the base function \(f(x)\). When we replace \(x\) with \(x-h\), where \(h\) is a positive real number, it results in shifting the graph \(h\) units to the right. Conversely, if we replace \(x\) with \(x+h\), the graph shifts \(h\) units to the left.

An example from the exercise is the exponential function \(f(x) = e^{-x}\), which when shifted 4 units right, becomes \(g(x) = e^{-(x-4)}\). This reflects each point on the graph of \(f\) being moved 4 units to the right to create the graph of \(g\).

Vertical Shift in Function

A vertical shift in a function happens when the entire graph moves up or down. This transformation affects the \(y\) value of each point in the graph.

To implement a vertical shift, we add or subtract a constant to the function. If we add a constant \(k\), the function \(f(x)\) moves up by \(k\) units, creating a new function \(g(x) = f(x) + k\). If we subtract \(k\), the graph shifts down by \(k\) units, leading to \(g(x) = f(x) - k\).

In the given exercise, after applying the horizontal shift to \(e^{-x}\), we subtract 1 to implement a downward shift. Hence \(g_1(x) = e^{-x+4} - 1\) represents a \(f\) translated 1 unit down.

Vertical Shrink in Function

A vertical shrink occurs when each point on the function's graph is moved closer to the \(x\)-axis, compressing the graph vertically.

Mathematically, vertical shrinking is achieved by multiplying the function by a fraction less than one. If we start with \(f(x)\) and multiply it by a factor of \(a\) where \(0 < a < 1\), we get the function \(g(x) = af(x)\), which is a vertically shrunk version of \(f(x)\).

In the exercise, the final transformation is a vertical shrink of \(e^{-x+4} - 1\) by a factor of \(\frac{1}{3}\), resulting in the function \(g(x) = \frac{1}{3}(e^{-x+4} - 1)\), squeezing the graph towards the \(x\)-axis.

Exponential Functions

An exponential function can be recognized by its use of a constant base raised to a variable exponent. Common bases are \(e\), the natural base, or 10, among others.

Exponential functions like \(f(x) = e^{-x}\) have distinct characteristics such as rapid growth or decay, and they are ubiquitous in fields like finance, computer science, and natural sciences. They often model phenomena where growth or decay accelerates over time.

Transformations applied to exponential functions, such as horizontal shifts, vertical shifts, and vertical shrinks, follow the same rules as polynomial functions. However, due to the nature of exponential growth or decay, the visual impact of these transformations can be quite different, especially on a graph's curvature and rate of change.