Graphing Exponential Functions and Properties

Graphing Exponential Functions

• Basic Form: ( f(x) = a \cdot b^x )
  • ( a ): vertical stretch/compression.
  • ( b ): base of the exponential (if ( b > 1 ), growth; if ( 0 < b < 1 ), decay).
• Key Features:
  • Y-Intercept: ( (0, a) ).
  • Horizontal Asymptote: ( y = 0 ) (as ( x \to -\infty )).
  • Increasing/Decreasing:
    • Increasing if ( b > 1 ).
    • Decreasing if ( 0 < b < 1 ).
• Domain & Range:
  • Domain: ( (-\infty, \infty) ).
  • Range: ( (0, \infty) ) for ( a > 0 ) or ( (-\infty, 0) ) for ( a < 0 ).

Properties of Exponents

1. Product of Powers: ( a^m \cdot a^n = a^{m+n} )
2. Quotient of Powers: ( a^m / a^n = a^{m-n} )
3. Power of a Power: ( (a^m)^n = a^{m \cdot n} )
4. Power of a Product: ( (ab)^n = a^n \cdot b^n )
5. Power of a Quotient: ( (a/b)^n = a^n / b^n )
6. Zero Exponent: ( a^0 = 1 ) (where ( a \neq 0 ))
7. Negative Exponent: ( a^{-n} = 1/a^n )

Logarithmic Identities

1. Definition: If ( b^y = x ), then ( \log_b(x) = y ).
2. Logarithm of a Product: ( \log_b(xy) = \log_b(x) + \log_b(y) )
3. Logarithm of a Quotient: ( \log_b(x/y) = \log_b(x) - \log_b(y) )
4. Logarithm of a Power: ( \log_b(x^n) = n \cdot \log_b(x) )
5. Change of Base Formula: ( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} )
6. Logarithm of 1: ( \log_b(1) = 0 )
7. Logarithm of the Base: ( \log_b(b) = 1 )

Exponential Growth and Decay

• Exponential Growth:
  • Model: ( P(t) = P_0 \cdot e^{rt} )
  • ( P_0 ): initial amount, ( r ): growth rate, ( t ): time.
• Exponential Decay:
  • Model: ( P(t) = P_0 \cdot e^{-rt} )
  • Characterized by a constant percentage decrease.
• Half-Life:
  • Time required for a quantity to reduce to half its initial value, commonly used in radioactive decay.

Applications of Logarithms

• Solving Exponential Equations: Use logarithmic identities to isolate the variable.
• pH Scale: ( \text{pH} = -\log_{10}[\text{H}^+] ) measures acidity.
• Richter Scale: Measures earthquake intensity logarithmically.
• Sound Intensity: Decibels also use logarithmic scales: ( dB = 10 \log_{10}(I/I_0) ).
• Finance: Compound interest calculations can involve logarithmic functions for time in growth models.

Graphing Exponential Functions

• Basic form of an exponential function: ( f(x) = a \cdot b^x ).
• Coefficient ( a ) indicates the vertical stretch or compression of the graph.
• Base ( b ) determines the nature of the function:
  • If ( b > 1 ), the function exhibits exponential growth.
  • If ( 0 < b < 1 ), it shows exponential decay.
• Key Features:
  • Y-intercept is located at ( (0, a) ).
  • The horizontal asymptote is established at ( y = 0 ) as ( x \to -\infty ).
  • The function is increasing for ( b > 1 ) and decreasing for ( 0 < b < 1 ).
• Domain & Range:
  • Domain covers all real numbers: ( (-\infty, \infty) ).
  • Range is ( (0, \infty) ) if ( a > 0 ) and ( (-\infty, 0) ) if ( a < 0 ).

Properties of Exponents

• Product of Powers: Multiply bases with the same exponent: ( a^m \cdot a^n = a^{m+n} ).
• Quotient of Powers: Subtract exponents when dividing same bases: ( a^m / a^n = a^{m-n} ).
• Power of a Power: Multiply exponents: ( (a^m)^n = a^{m \cdot n} ).
• Power of a Product: Distribute exponent over multiplication: ( (ab)^n = a^n \cdot b^n ).
• Power of a Quotient: Distribute exponent over division: ( (a/b)^n = a^n / b^n ).
• Zero Exponent: Any non-zero base raised to the power of zero equals one: ( a^0 = 1 ).
• Negative Exponent: Inverse of the base raised to a positive power: ( a^{-n} = 1/a^n ).

Logarithmic Identities

• Definition: Establishes the inverse relationship between exponentiation and logarithms: if ( b^y = x ), then ( \log_b(x) = y ).
• Logarithm of a Product: Sum of logarithms when multiplying: ( \log_b(xy) = \log_b(x) + \log_b(y) ).
• Logarithm of a Quotient: Difference of logarithms when dividing: ( \log_b(x/y) = \log_b(x) - \log_b(y) ).
• Logarithm of a Power: Coefficient in front when exponentiating: ( \log_b(x^n) = n \cdot \log_b(x) ).
• Change of Base Formula: Converts one logarithmic base to another: ( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} ).
• Logarithm of 1: Always equals zero: ( \log_b(1) = 0 ).
• Logarithm of the Base: Always equals one: ( \log_b(b) = 1 ).

Exponential Growth and Decay

• Exponential Growth Model: Describes growth over time: ( P(t) = P_0 \cdot e^{rt} ), where ( P_0 ) is the initial amount, ( r ) is the growth rate, and ( t ) represents time.
• Exponential Decay Model: Describes a decrease: ( P(t) = P_0 \cdot e^{-rt} ), characterized by a consistent percentage decrease.
• Half-Life: Refers to the time needed for a quantity to reduce to half of its original amount, particularly important in radioactive decay.

Applications of Logarithms

• Solving Exponential Equations: Employ logarithmic identities to isolate the variable effectively.
• pH Scale: Measures hydrogen ion concentration in acidity, defined as ( \text{pH} = -\log_{10}[\text{H}^+] ).
• Richter Scale: Utilizes a logarithmic scale to measure the intensity of earthquakes.
• Sound Intensity: Expressed in decibels, calculated using: ( dB = 10 \log_{10}(I/I_0) ), where ( I_0 ) is a reference intensity.
• Finance: Logarithmic functions facilitate calculations involving compound interest, particularly in modeling growth over time.

This quiz covers the fundamental concepts of graphing exponential functions and the properties of exponents. Learn about the basic form of exponential functions, key features like y-intercepts and asymptotes, as well as important exponent rules. Test your knowledge and understanding of these crucial mathematical concepts.