Question

the equation for h(x) and the transformed function p(x) in terms of h(x) are shown. which statement describes the transformation(s) performed on h(x) that produced p(x)?
h(x) = x³
p(x) = 2h(x + 4) + 5
a stretched vertically by a factor of 2; shifted left 4 units; shifted up 5 units
b stretched vertically by a factor of 2; shifted right 4 units; shifted down 5 units
c stretched vertically by a factor of 4; shifted right 4 units; shifted down 6 units
d stretched vertically by a factor of 4; shifted left 2 units; shifted up 5 units

Explanation:

Step1: Analyze Vertical Stretch

For a function \( y = a \cdot h(x) \), if \( |a|>1 \), it's a vertical stretch by factor \( |a| \). Here, \( p(x)=2h(x + 4)+5 \), so \( a = 2 \), vertical stretch by 2.

Step2: Analyze Horizontal Shift

For \( y = h(x + k) \), if \( k>0 \), shift left \( k \) units. Here, \( x+4 \) means \( k = 4 \), shift left 4 units.

Step3: Analyze Vertical Shift

For \( y = h(x)+k \), if \( k>0 \), shift up \( k \) units. Here, \( +5 \) means shift up 5 units.

Answer:

A. Stretched vertically by a factor of 2; shifted left 4 units; shifted up 5 units