MCR3U – Grade 11 Functions – Exponential Functions Test Notes

Exponential Unit Study Notes

Exponent Laws

Exponent Laws Formula
Multiplication (a^n)*(a^n) = a^(m+n)
Division (a^n)/(a^m) = a^(n-m)
Power (a^n)^m = a^(n*m)
Negative A^-b = (1/a)^b
Zero A^0 = 1
Power of a product (xy)^a = x^a*y^a
Power of a quotient (x/y)^a = (x^a)/(y^a)
Fractional Exponent A^(2/2) = 2 root (a^2)

 

Basic Techniques

-Take multiplying terms with negative exponents down or up in a fraction to remove negativity

-Terms with the same base can have exponent’s added/subtracted/multiplied

-Solving variables in the exponent with same base can eliminate the base so that you only need to deal solving the exponent variables.

Exponential Functions

-f(x) = ab^(x)

-b : is the base of the exponent

-if b > 0, then the function is increasing

-if b < 0, then the function is decreasing

-A is the initial value (y intercept) which also defines the asymptote

Transformations of functions

            Vertical Stretch

-a’s value dictates vertical stretch/compression

-if a > 1, then the function is stretched vertically by the factor of a

-if 0 < a < 1, then the function is compressed by the factor of a

-Values of y increases/decreases per x, x doesn’t change

            Vertical Reflection

-if a < 0, then the function is reflected off the x axis

            Vertical Translation

            –y = ab^(x)+k

-K dictates the vertical translation up or down

-If K is positive, the entire translation is shifted up by the value of k

-If K is negative, the entire translation is shifted down by the value of k

Horizontal Translation

            -y = ab^(x+k)

-K dictates the horizontal translation left and right

-if K is negative, the function shifts right by the value of k

– if K is positive, the function shifts left by the value of k

-sometimes, the true translation/value of K be revealed if the horizontal stretch is factored

Horizontal Stretch

  • y = ab^p(x)
  • P dictates the horizontal stretch and compression
  • P should be factored out of the bracket
  • If P > 1, then the function is compressed by the factor of 1/p
  • If 0 > p > 1, then the function is stretched by the factor of 1/p
  • Stretch factor is the reciprocal of what’s displayed in the equation

Horizontal Reflection

  • The value of P from horizontal stretching defines the reflection
  • If it’s negative, the function will reflect off the Y axis.