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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblema 63
Capitolo 5, Problema 63

Give the equation of each exponential function whose graph is shown.
Graph of a decreasing exponential function passing through points (0, -1), (1, -e), and (2, -e squared).

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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.
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 \(a = 4\).
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 \(b\).
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 \(\frac{1}{e} = e^{-4b}\).
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 \(b\).

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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.
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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.
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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.
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