Axes of symmetry?

Question

What is the equations for the axes of symmetry for the absolute functions of y=absolute value of x, y=absolute value of x plus 2, and y=absolute value of (x-3)+2?

Answer 1

What is the equations for the axes of symmetry for the absolute functions of y = |x|, y= |x| + 2 OR y = |x + 2| and y = |x - 3| + 2?

i. y = |x|
ie y = x for x ≥ 0
= -x for x < 0

So the bottom of the V is at the origin
ie x = 0 is the axis of symmetry

ii. y= |x| + 2
ie y = x + 2 for x ≥ 0
= -x + 2 for x < 0

The bottom of the V this time is at (0, 2)
So x = 0 is the axis of symmetry

OR y = |x + 2|

ie y = x + 2 for x + 2 ≥ 0 ie x ≥ -2
and = -(x + 2) for x + 2 < 0 ie x < -2

So the V here is at (-2, 0)
So x = -2 is the axis of symmetry

iii. y= |x - 3|+2

So y = (x - 3) + 2 for x - 3 ≥ 0
ie y = x - 1 for x ≥ 3
and y = - (x - 3) + 2 for x - 3 < 0
ie y = 5 - x for x < 3

In this case the bottom of the V is at (3, 2)
So x = 3 is the axis of symmetry

Answer 2

The first has an axis of symmetry x = 0.

If you add 2 outside the absolute value it just moves the graph up 2 so it would still have x=0. But if the 2 is inside the absolute value it moves the graph 2 to the left so the axis shifts to x = -2.

If the -3 is inside the absolute value, the graph moves 3 to the right so its axis would be x = 3.

Answer 3

x=0

Answer 4

y = absolute value of x : x = 0
y = absolute value of x plus 2: x = 0
y = absolute value of (x-3) + 2 x= 0

you can use the value table to find out why the answer is x = 0

Answer 5

Sketch them and take a look.