Question

exponential functions
parent
( y = 3^x )
( y = 5^x )
stretch
shrink
reflection
( y = 0.5(3)^x )
( y = 2(3)^x )
( y = 5 cdot 5^x )
( y = -5^x )
( y = -2(3)^x )
( y = 0.1(5)^x )

Explanation:

Step1: Identify Parent Functions

The parent exponential functions are of the form \( y = a^x \) where \( a>0,a
eq1 \) and there is no vertical transformation (coefficient 1). So \( y = 3^x \) (base 3) and \( y = 5^x \) (base 5) are parent functions.

Step2: Identify Stretch Functions

A vertical stretch of a function \( y = f(x) \) is given by \( y = kf(x) \) where \( k>1 \). For \( y = 2(3)^x \), \( k = 2>1 \) (stretch of \( y = 3^x \)). For \( y = 5\cdot5^x \), \( k = 5>1 \) (stretch of \( y = 5^x \)).

Step3: Identify Shrink Functions

A vertical shrink of a function \( y = f(x) \) is given by \( y = kf(x) \) where \( 0

Step4: Identify Reflection Functions

A reflection over the x - axis of a function \( y = f(x) \) is given by \( y=-f(x) \). For \( y=-5^x \), it is a reflection of \( y = 5^x \) over the x - axis. For \( y=-2(3)^x \), it is a reflection of \( y = 2(3)^x \) (or \( y = 3^x \)) over the x - axis (the negative sign causes the reflection).

Answer:

Parent:

• \( y = 3^x \), \( y = 5^x \)

Stretch:

• \( y = 2(3)^x \), \( y = 5\cdot5^x \) (Note: \( y = 5\cdot5^x = 5^{x + 1} \), but for the purpose of vertical stretch, the coefficient greater than 1: \( y = 2(3)^x \) (coefficient 2 > 1) and \( y = 5\cdot5^x \) (coefficient 5 > 1))

Shrink:

• \( y = 0.5(3)^x \), \( y = 0.1(5)^x \) (Coefficients \( 0.5 \) and \( 0.1 \) are between 0 and 1, so they represent vertical shrink)

Reflection:

• \( y = -5^x \), \( y = -2(3)^x \) (Negative coefficient indicates reflection over the x - axis)