Chapter 5: Exponential Functions

Question 1

Convert rational exponents (power is a fraction) to a surd

Answer 1

a¹/ⁿ = n√ a
aᵐ/ⁿ = n√ aᵐ

Question 2

Define and solve exponential equations
(Eg. 3ˣ⁺² = 1/27)

Answer 2

Exponential equation: equation where unknown occurs as part of the power
- Can be solved by expressing both exponents to have the same base
(aᵐ=aⁿ, m=n)
[Use rules of exponents and other algebraic rules to form identical bases)

Eg.
3ˣ⁺² = 1/27
3ˣ⁺² = 1/3³
3ˣ⁺² = 3⁻³
x+2 = -3
x = -5

Question 3

Draw and state the properties of an exponential graph* (y=aˣ, a>0, a≠1)

Answer 3

1) Range is y>0 (graph is always positive)
(aˣ>0 when a>0)
2) Horizontal asymptote is y=0
(aˣ≠0 when a>0 )
3) y-intercept is 1
(x=0, a⁰=1)

Question 4

Translate exponential graphs
(p x aᑫˣ⁻ʰ + k)

Answer 4

• h controls the horizontal translation
• k controls the vertical translation
  (Horizontal asymptote is also y=k)
• p controls the vertical stretch
• q controls the horizontal stretch
• Sign of p controls the reflection at the x-axis
• Sign of x controls the reflection at the y-axis