Question

amples
write the equation of the function with the given transformations.

1. quadratic function; translated right 4, down 7
2. absolute value function; translated up 10
3. use the table below to identify the transformations and write the equation of the absolute value function f(x).

x	-6	-5	-4	-3	-2
f(x)	3	1	-1	1	3

1. use the table below to identify the transformations and write the equation of the quadratic function f(x).

x	0	1	2	3	4
f(x)	7	4	3	4	7

Explanation:

Problem 5:

Step1: Recall parent quadratic function

The parent quadratic function is \( y = x^2 \).

Step2: Apply horizontal translation

A translation right 4 units means replacing \( x \) with \( x - 4 \), so we get \( y=(x - 4)^2 \).

Step3: Apply vertical translation

A translation down 7 units means subtracting 7 from the function, so the equation becomes \( y=(x - 4)^2-7 \).

Step1: Recall parent absolute value function

The parent absolute value function is \( y = |x| \).

Step2: Apply vertical translation

A translation up 10 units means adding 10 to the function, so the equation becomes \( y = |x|+10 \).

Step1: Find vertex of absolute value function

For an absolute value function, the vertex is the minimum (or maximum) point. Looking at the table, when \( x=-4 \), \( f(x)=-1 \), and the values are symmetric around \( x = - 4 \) (since \( f(-6)=f(-2)=3 \), \( f(-5)=f(-3)=1 \)). So the vertex is \( (-4,-1) \).

Step2: Recall vertex form of absolute value function

The vertex form of an absolute value function is \( y=a|x - h|+k \), where \( (h,k) \) is the vertex. Here \( h=-4 \), \( k = - 1 \), so \( y=a|x+4|-1 \).

Step3: Find the value of \( a \)

Use a point, say \( x=-5 \), \( f(x)=1 \). Substitute into the equation: \( 1=a|-5 + 4|-1 \), \( 1=a|-1|-1 \), \( 1=a(1)-1 \), \( a=2 \).

Step4: Write the equation

Substitute \( a = 2 \), \( h=-4 \), \( k=-1 \) into the vertex form: \( y = 2|x + 4|-1 \).

Answer:

\( y=(x - 4)^2-7 \)

Problem 6: