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Geometric Sequences & Exponential Functions

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Geometric Sequences & Exponential Functions Algebra I GEOMETRIC SEQUENCES Study Guides Big Picture & EXPONENTIAL FUNCTIONS Both geometric sequences and exponential functions serve as a way to represent the repeated and patterned multiplication of numbers and variables. Exponential functions can be used to represent things seen in the natural world, such as population growth or compound interest in a bank. Key Terms Geometric Sequence: A sequence of numbers in which each number in the sequence is found by multiplying the previous number by a fixed amount called the common ratio. Exponential Function: A function with the form y = A ∙ bx. Geometric Sequences A geometric sequence is a type of pattern where every number in the sequence is multiplied by a certain number called the common ratio. Example: 4, 16, 64, 256, ... Find the common ratio r by dividing each term in the sequence by the term before it. So r = 4 Exponential Functions An exponential function is like a geometric sequence, except geometric sequences are discrete (can only have values at certain points, e.g. 4, 16, 64, 256, ...) and exponential functions are continuous (can take on all possible values). Exponential functions look like y = A ∙ bx, where A is the starting amount and b is like the common ratio of a geometric sequence. Graphing Exponential Functions Here are some examples of exponential functions: Exponential Growth and Decay Growth: y = A ∙ bx when b ≥ 1 Decay: y = A ∙ bx when b is between 0 and 1 your textbook and is for classroom or individual use only. your Disclaimer: this study guide was not created to replace Disclaimer: this study guide was • In exponential growth, the value of y increases (grows) as x increases. See the blue curve. • In exponential decay, the value of y decreases (de- cays) as x increases. See the red curve. This guide was created by Nicole Crawford, Jane Li, and Jin Yu. To learn more Page 1 of 2 about the student authors, visit http://www.ck12.org/about/ck-12-interns/. v1.1.9.2012 GEOMETRIC SEQUENCES Logarithms & EXPONENTIAL FUNCTIONS CONT. To undo an exponential function, we can take the logarithm of the equation. y • logb x = y means that b = x y • log x = log10 x = y means that 10 = x Algebra I Comparing Linear, Quadratic, and Exponential Models • Linear function: y = mx + b • Quadratic function: y = ax2 + bx + c • Exponential function: y = a ∙ bx Identifying Linear Models If the difference between values of the output (dependent variable, usually the y-variable) is the same each time we change the independent variable (input, usually the x-variable) by the same amount, then the function is linear. Identifying Quadratic Models The difference between the x-values is constant, but the difference between the y-values is not. However, there is a constant interval between the y-values. Identifying Exponential Models If the ratio between values of the dependent variable (output, y-variable) has stayed the same each time we change the independent variable (input, x-variable) by the same amount, then the function is exponential. Notes Page 2 of 2.
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