Exponent Rules Parts n When a number variable
Exponent Rules
Parts n 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? n n An exponent means that you multiply the base by itself that many times. For example x 4 = x ● x ● x 26 = 2 ● 2 ● 2 ● 2 = 64
The Invisible Exponent n When an expression does not have a visible exponent its exponent is understood to be 1.
Exponent Rule #1 n n When multiplying two expressions with the same base you add their exponents. For example
Exponent Rule #1 n Try it on your own:
Exponent Rule #2 n n When dividing two expressions with the same base you subtract their exponents. For example
Exponent Rule #2 n Try it on your own:
Exponent Rule #3 n n When raising a power to a power you multiply the exponents For example
Exponent Rule #3 n Try it on your own
Note n 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 n n When a product is raised to a power, each piece is raised to the power For example
Exponent Rule #4 n Try it on your own
Note n 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 n n When a quotient is raised to a power, both the numerator and denominator are raised to the power For example
Exponent Rule #5 n Try it on your own
Zero Exponent n When anything, except 0, is raised to the zero power it is 1. ( if a ≠ 0) n For example ( if x ≠ 0)
Zero Exponent ( if a ≠ 0) n Try it on your own ( if h ≠ 0)
Negative Exponents n If b ≠ 0, then n For example
Negative Exponents n If b ≠ 0, then n Try it on your own:
Negative Exponents n The negative exponent basically flips the part with the negative exponent to the other half of the fraction.
Math Manners n For a problem to be completely simplified there should not be any negative exponents
Mixed Practice
Mixed Practice
Mixed Practice
Mixed Practice
Mixed Practice F O I L
Mixed Practice
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) = 3 x and g(x) = 0. 5 x are exponential functions.
The value of f(x) = 3 x when x = 2 is f(2) = 32 = 9 The value of f(x) = 3 x when x = – 2 is f(– 2) = 3– 2 = The value of g(x) = 0. 5 x when x = 4 is g(4) = 0. 54 = 0. 0625
The graph of f(x) = ax, a > 1 Exponential Growth Function y 4 Range: (0, ) (0, 1) x 4 Domain: (– , ) Horizontal Asymptote y=0
The graph of f(x) = ax, 0 < a < 1 y Exponential Decay Function 4 Range: (0, ) (0, 1) x 4 Domain: (– , ) Horizontal Asymptote y=0
Exponential Function n n 3 Key Parts 1. Pivot Point (Common Point) 2. Horizontal Asymptote 3. Growth or Decay
Manual Graphing n Lets graph the following together: n f(x) = 2 x Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Example: Sketch the graph of f(x) = 2 x. x -2 -1 0 1 2 f(x) (x, f(x)) ¼ (-2, ¼) ½ (-1, ½) 1 (0, 1) 2 (1, 2) 4 (2, 4) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. y 4 2 x – 2 2
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) = 2 x g(x) = 10 x h(x) = 3 x Base is 2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Base is 10. Base is 3.
Calculator Comparison n n Graph the following on your calculator at the same time and note the trend y 1 = 2 x y 2= 5 x y 3 = 10 x
When base is a fraction n n Graph the following on your calculator at the same time and note the trend y 1 = (1/2)x y 2= (3/4)x y 3 = (7/8)x
Transformations Involving Exponential Functions Transformation Equation Description Horizontal translation g(x) = bx+c • Shifts Vertical stretching or shrinking g(x) = cbx 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. Reflecting g(x) = -bx g(x) = b-x • Reflects Vertical translation g(x) = bx+ c • 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. the graph of f (x) = bx about the x-axis. • Reflects the graph of f (x) = bx about the y-axis. 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.
Example: Sketch the graph of g(x) = 2 x – 1. State the domain and range. The graph of this function is a vertical translation of the graph of f(x) = 2 x down one unit. y f(x) = 2 x 4 2 Domain: (– , ) x Range: (– 1, ) y = – 1
Example: Sketch the graph of g(x) = 2 -x. State the domain and range. y The graph of this function is a reflection the graph of f(x) = 2 x in the yaxis. f(x) = 2 x 4 Domain: (– , ) Range: (0, ) x – 2 2
Discuss these transformations n n n y = 2(x+1) Left 1 unit y = 2 x + 2 Up 2 units y = 2 -x – 2 Ry, then down 2 units
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