Exponent Rules

Parts When a number, variable, or expression is raised to a power, the number, variable, or expression is called the base and the power is called the exponent.

What is an Exponent? x4 = x ● x ● x ● x 26 = 2 ● 2 ● 2 ● 2 ● 2 ● 2 An exponent means that you multiply the base by itself that many times. For example x4 = x ● x ● x ● x 26 = 2 ● 2 ● 2 ● 2 ● 2 ● 2 = 64

The Invisible Exponent When an expression does not have a visible exponent its exponent is understood to be 1.

Exponent Rule #1 When multiplying two expressions with the same base you add their exponents. For example

Exponent Rule #1 Try it on your own:

Exponent Rule #2 When dividing two expressions with the same base you subtract their exponents. For example

Exponent Rule #2 Try it on your own:

Exponent Rule #3 When raising a power to a power you multiply the exponents For example

Exponent Rule #3 Try it on your own

Note When using this rule the exponent can not be brought in the parenthesis if there is addition or subtraction You would have to use FOIL in these cases

Exponent Rule #4 When a product is raised to a power, each piece is raised to the power For example

Exponent Rule #4 Try it on your own

Note This rule is for products only. When using this rule the exponent can not be brought in the parenthesis if there is addition or subtraction You would have to use FOIL in these cases

Exponent Rule #5 When a quotient is raised to a power, both the numerator and denominator are raised to the power For example

Exponent Rule #5 Try it on your own

Zero Exponent When anything, except 0, is raised to the zero power it is 1. For example ( if a ≠ 0) ( if x ≠ 0)

Zero Exponent Try it on your own ( if a ≠ 0) ( if h ≠ 0)

Negative Exponents If b ≠ 0, then For example

Negative Exponents If b ≠ 0, then Try it on your own:

Negative Exponents The negative exponent basically flips the part with the negative exponent to the other half of the fraction.

Math Manners For a problem to be completely simplified there should not be any negative exponents

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Definition of Exponential Function The exponential function f with base a is defined by f(x) = ax where a > 0, a  1, and x is any real number. For instance, f(x) = 3x and g(x) = 0.5x are exponential functions. Definition of Exponential Function

Example: Exponential Function The value of f(x) = 3x when x = 2 is f(2) = 32 = 9 The value of f(x) = 3x when x = –2 is f(–2) = 3–2 = The value of g(x) = 0.5x when x = 4 is g(4) = 0.54 = 0.0625 Example: Exponential Function

Graph of Exponential Function (a > 1) The graph of f(x) = ax, a > 1 y Exponential Growth Function 4 Range: (0, ) (0, 1) x 4 Horizontal Asymptote y = 0 Domain: (–, ) Graph of Exponential Function (a > 1)

Graph of Exponential Function (0 < a < 1) The graph of f(x) = ax, 0 < a < 1 y Exponential Decay Function 4 Range: (0, ) (0, 1) x 4 Horizontal Asymptote y = 0 Domain: (–, ) Graph of Exponential Function (0 < a < 1)

Exponential Function 3 Key Parts 1. Pivot Point (Common Point) 2. Horizontal Asymptote 3. Growth or Decay

Manual Graphing Lets graph the following together: f(x) = 2x Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Example: Sketch the graph of f(x) = 2x. x f(x) (x, f(x)) y x f(x) (x, f(x)) -2 ¼ (-2, ¼) -1 ½ (-1, ½) 1 (0, 1) 2 (1, 2) 4 (2, 4) 4 2 x –2 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Graph f(x) = 2x

Definition of the Exponential Function The exponential function f with base b is defined by f (x) = bx or y = bx Where b is a positive constant other than and x is any real number. Here are some examples of exponential functions. f (x) = 2x g(x) = 10x h(x) = 3x Base is 2. Base is 10. Base is 3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Calculator Comparison Graph the following on your calculator at the same time and note the trend y1 = 2x y2= 5x y3 = 10x

When base is a fraction Graph the following on your calculator at the same time and note the trend y1 = (1/2)x y2= (3/4)x y3 = (7/8)x

Transformations Involving Exponential Functions Shifts the graph of f (x) = bx upward c units if c > 0. Shifts the graph of f (x) = bx downward c units if c < 0. g(x) = bx+ c Vertical translation Reflects the graph of f (x) = bx about the x-axis. Reflects the graph of f (x) = bx about the y-axis. g(x) = -bx g(x) = b-x Reflecting Multiplying y-coordintates of f (x) = bx by c, Stretches the graph of f (x) = bx if c > 1. Shrinks the graph of f (x) = bx if 0 < c < 1. g(x) = cbx Vertical stretching or shrinking Shifts the graph of f (x) = bx to the left c units if c > 0. Shifts the graph of f (x) = bx to the right c units if c < 0. g(x) = bx+c Horizontal translation Description Equation Transformation

Example: Translation of Graph Example: Sketch the graph of g(x) = 2x – 1. State the domain and range. y f(x) = 2x The graph of this function is a vertical translation of the graph of f(x) = 2x down one unit . 4 2 Domain: (–, ) x y = –1 Range: (–1, ) Example: Translation of Graph

Example: Reflection of Graph Example: Sketch the graph of g(x) = 2-x. State the domain and range. y f(x) = 2x The graph of this function is a reflection the graph of f(x) = 2x in the y-axis. 4 Domain: (–, ) x –2 2 Range: (0, ) Example: Reflection of Graph

Discuss these transformations y = 2(x+1) Left 1 unit y = 2x + 2 Up 2 units y = 2-x – 2 Ry, then down 2 units