Question

warm-up
each function below is a translation of the graph of the function ( f(x) = x^2 ). match the function with the description of its relationship to the parent function.
( m(x) = (x - 1)^2 )
( k(x) = x^2 + 1 )
( h(x) = x^2 - 1 )
( g(x) = (x + 1)^2 )
down 1
left 1
right 1
up 1

Explanation:

Step1: Recall translation rules

For a function \( y = f(x - h) + k \), horizontal translation: \( h>0 \) shifts right \( h \) units, \( h<0 \) shifts left \( |h| \) units. Vertical translation: \( k>0 \) shifts up \( k \) units, \( k<0 \) shifts down \( |k| \) units.

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

Compare with \( f(x)=x^2 \), here \( h = 1 \), \( k = 0 \). So horizontal shift right 1 unit.

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

Here \( h = 0 \), \( k = 1 \). So vertical shift up 1 unit.

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

Here \( h = 0 \), \( k=- 1 \). So vertical shift down 1 unit.

Step5: Analyze \( g(x)=(x + 1)^2=(x-(-1))^2 \)

Here \( h=-1 \), \( k = 0 \). So horizontal shift left 1 unit.

Answer:

• \( m(x)=(x - 1)^2 \) → right 1
• \( k(x)=x^2 + 1 \) → up 1
• \( h(x)=x^2 - 1 \) → down 1
• \( g(x)=(x + 1)^2 \) → left 1