Question

match the quadratics with their transformations. (how did they move?)
$f(x) = (x - 1)^2$
$f(x) = x^2 + 1$
$f(x) = (x + 1)^2 + 1$
$f(x) = (x - 1)^2 + 1$
$f(x) = x^2$

shifted right 1 & up 1
shifted right 1
shifted left 1 & up 1
parent function
shifted up 1

Explanation:

Step1: Analyze \( f(x)=(x - 1)^2 \)

The parent function is \( f(x)=x^2 \). For a function \( f(x)=(x - h)^2 \), it represents a horizontal shift of \( h \) units. Here \( h = 1 \), so it's shifted right 1.

Step2: Analyze \( f(x)=x^2 + 1 \)

For a function \( f(x)=x^2 + k \), \( k = 1 \) means it's shifted up 1 unit from the parent function \( f(x)=x^2 \).

Step3: Analyze \( f(x)=(x + 1)^2 + 1 \)

\( (x + 1)^2 \) is a shift left 1 (since \( h=- 1\) in \( (x - h)^2 \)) and \( +1 \) is a shift up 1, so shifted left 1 & up 1.

Step4: Analyze \( f(x)=(x - 1)^2 + 1 \)

\( (x - 1)^2 \) is shift right 1 and \( +1 \) is shift up 1, so shifted right 1 & up 1.

Step5: Analyze \( f(x)=x^2 \)

This is the parent function, so it matches "Parent function".

Answer:

• \( f(x)=(x - 1)^2 \) - Shifted right 1
• \( f(x)=x^2 + 1 \) - Shifted up 1
• \( f(x)=(x + 1)^2 + 1 \) - Shifted left 1 & up 1
• \( f(x)=(x - 1)^2 + 1 \) - Shifted right 1 & up 1
• \( f(x)=x^2 \) - Parent function