Give the equation of each exponential function whose graph is shown.

Verified step by step guidance

1. Step 1: Identify the general form of the exponential function. Since the graph shows an exponential function with negative values, it likely has the form \(y = -a e^{bx}\), where \(a > 0\) and \(b\) is a constant to be determined.
2. Step 2: Use the point where the graph crosses the y-axis, which is \((0, -4)\). Substitute \(x=0\) and \(y=-4\) into the equation \(y = -a e^{bx}\) to find \(a\). Since \(e^{0} = 1\), this gives \(-4 = -a \cdot 1\), so \(a = 4\).
3. Step 3: Now the equation is \(y = -4 e^{bx}\). Use another point from the graph, for example \((-4, -\frac{4}{e})\), and substitute \(x = -4\) and \(y = -\frac{4}{e}\) into the equation to solve for \(b\).
4. Step 4: Substitute the values into the equation: \(-\frac{4}{e} = -4 e^{b(-4)}\). Simplify this to \(\frac{4}{e} = 4 e^{-4b}\), then divide both sides by 4 to get \(\frac{1}{e} = e^{-4b}\).
5. Step 5: Recognize that \(\frac{1}{e} = e^{-1}\), so set \(e^{-1} = e^{-4b}\). Since the bases are the same, equate the exponents: \(-1 = -4b\), and solve for \(b\).

Key Concepts

Exponential Function Form

An exponential function is generally written as y = ab^x, where a is the initial value (y-intercept) and b is the base that determines the growth or decay rate. Understanding this form helps in identifying the function from given points on its graph.

Using Points to Find Parameters

Given points on the graph, you can substitute their coordinates into the exponential function to create equations. Solving these equations allows you to find the values of a and b, which define the specific exponential function.

Properties of the Number e

The number e (~2.718) is a special base for exponential functions, often used in continuous growth or decay models. Recognizing e in the points (like -4/e or -4e) helps in simplifying and understanding the function's behavior.