Question

recall your knowledge from previous lessons. use words to describe the transformations on the quadratic parent function, $f(x)=x^2$.

1. $f(x)=(x - 1)^2$
2. $f(x)=(x + 1)^2$
3. $f(x)=x^2 + 1$
4. $f(x)=x^2 - 1$

Explanation:

1. For \( f(x) = (x - 1)^2 \)

Step1: Recall horizontal shift rule

For a function \( y = f(x - h) \), if \( h>0 \), it's a shift right by \( h \) units from \( y = f(x) \).
Here, \( f(x)=(x - 1)^2 \) and parent function is \( f(x)=x^2 \), so \( h = 1>0 \).

Step2: Describe the transformation

The graph of \( f(x)=(x - 1)^2 \) is the graph of the parent function \( f(x)=x^2 \) shifted 1 unit to the right.

Step1: Recall horizontal shift rule

For a function \( y = f(x + h) \), if \( h>0 \), it's a shift left by \( h \) units from \( y = f(x) \). Rewrite \( (x + 1)^2 \) as \( (x-(- 1))^2 \), so \( h = 1>0 \).

Step2: Describe the transformation

The graph of \( f(x)=(x + 1)^2 \) is the graph of the parent function \( f(x)=x^2 \) shifted 1 unit to the left.

Step1: Recall vertical shift rule

For a function \( y = f(x)+k \), if \( k>0 \), it's a shift up by \( k \) units from \( y = f(x) \). Here, \( k = 1>0 \).

Step2: Describe the transformation

The graph of \( f(x)=x^2 + 1 \) is the graph of the parent function \( f(x)=x^2 \) shifted 1 unit up.

Answer:

The graph of \( f(x)=(x - 1)^2 \) is the graph of \( f(x)=x^2 \) shifted 1 unit to the right.

2. For \( f(x) = (x + 1)^2 \)