Question

identify the graph of ( p(x) = 3(x - 1)^2 )
compare the graph to the graph of ( f(x) = x^2 )
the graph of ( p ) is a ( square ) down translation ( square ) units ( square ) and a vertical stretch by a factor of ( square ) of the graph of ( f )

Explanation:

Step1: Recall transformation rules for quadratic functions

For a quadratic function in the form \( p(x) = a(x - h)^2 + k \), compared to \( f(x)=x^2 \):

• The horizontal translation is determined by \( h \): if \( h>0 \), it's a shift to the right by \( h \) units; if \( h < 0 \), it's a shift to the left by \( |h| \) units.
• The vertical stretch/compression is determined by \( |a| \): if \( |a|>1 \), it's a vertical stretch by a factor of \( |a| \); if \( 0<|a|<1 \), it's a vertical compression.
• The vertical translation is determined by \( k \): if \( k>0 \), it's a shift up by \( k \) units; if \( k < 0 \), it's a shift down by \( |k| \) units.

Step2: Analyze \( p(x)=3(x - 1)^2 \)

Given \( p(x)=3(x - 1)^2 \) and \( f(x)=x^2 \).

• For horizontal translation: Comparing with \( a(x - h)^2 + k \), here \( h = 1 \) and \( k = 0 \). Since \( h=1>0 \), the graph of \( p(x) \) is a translation to the right by 1 unit (not down, the original "down" in the problem is incorrect; it should be "right").
• For vertical stretch: Here \( a = 3 \), and \( |3|>1 \), so it's a vertical stretch by a factor of 3.

Answer:

The graph of \( p \) is a \(\boldsymbol{\text{right}}\) translation \(\boldsymbol{1}\) units \(\boldsymbol{\text{right}}\) and a vertical stretch by a factor of \(\boldsymbol{3}\) of the graph of \( f \). (Note: The original "down" in the problem is an error; the horizontal translation for \( p(x)=3(x - 1)^2 \) relative to \( f(x)=x^2 \) is a shift to the right by 1 unit as \( h = 1 \) in the vertex form \( y=a(x - h)^2+k \))