Question

2.7 independent practice & hw

1. describe each transformation as it relates to the graph of the parent function. then, sketch its graph in relation to the parent function provided.

a. $y = x^2 + 2$
transformation(s):

b. $y = (x - 3)^2$
transformation(s):

d. $h(x) = (x + 1)^2 - 4$
transformation(s):

e. $h(x) = (x - 2)^2 + 3$
transformation(s):

g. $f(x) = - (x + 4)^2 + 1$
transformation(s):

h. $y = - (x - 1)^2 - 2$
transformation(s):

1. use the graph of each transformed parent function to describe the transformation and write its equation.

a.
transformation(s):
$f(x) = $

b.
transformation(s):
$f(x) = $

c.
transformation(s):
$f(x) = $

d.
transformation(s):
$f(x) = $

Explanation:

Part 1a: \( y = x^2 + 2 \)

Step1: Identify Parent Function

The parent function is \( y = x^2 \) (a parabola opening upwards with vertex at \((0,0)\)).

Step2: Analyze Vertical Shift

The given function is \( y = x^2 + 2 \). For a function \( y = f(x) + k \), if \( k>0 \), it shifts the graph of \( f(x) \) up by \( k \) units. Here, \( k = 2 \), so the graph of \( y = x^2 \) is shifted up 2 units.

Step1: Identify Parent Function

Parent function is \( y = x^2 \) (vertex at \((0,0)\)).

Step2: Analyze Horizontal Shift

For a function \( y = f(x - h) \), if \( h>0 \), it shifts the graph of \( f(x) \) to the right by \( h \) units. Here, \( h = 3 \), so the graph of \( y = x^2 \) is shifted right 3 units.

Step1: Identify Parent Function

Parent function: \( y = x^2 \) (vertex at \((0,0)\)).

Step2: Analyze Horizontal Shift

For \( y = (x + 1)^2 \), this is \( y = f(x - (-1)) \), so \( h=-1 \), which shifts the graph left 1 unit (since \( h<0 \) means left shift by \( |h| \) units).

Step3: Analyze Vertical Shift

Then, \( y = (x + 1)^2 - 4 \) is \( y = f(x) - 4 \), so it shifts the graph of \( (x + 1)^2 \) down by 4 units (since \( k=-4<0 \)).

Answer:

Transformation: The graph of the parent function \( y = x^2 \) is shifted vertically upward by 2 units.

Part 1b: \( y = (x - 3)^2 \)