Question

Please Help
Graph f(x) = -|x-1| +4 and determine the interval over which f(x) is an increasing function.
Write your answer in interval notation.

Answer 1

Do you know what the parent function \[f(x) = |x|\] looks like on the graph?

Answer 2

No

Answer 3

ok

Answer 4

Ask yourself this:
1) What happens to the graph if I multiply everything by -1?
2) What happens if I add 5 to every value? Do all of the points move up or down?

Answer 5

Multiplying everything by -1 will cause a reflection of the graph

Answer 6

Yes, so it will point down. so if you had the points

(1,1), (-1,1) they would become (1,-1) and (-1,-1).
What would happen to these points if you added 5 to f(x) ?

Answer 7

it will point up

Answer 8

(1,-1) will become (6,4) and (-1,-1) becomes (4,4)

Answer 9

sorry when you just add to the f(x) you don't add it to the x value.
so
(1,-1) would become (1,4) and
(-1,-1) would also become (1,4)

This mimics

f(x) = |x| parent function
g(x)=-|x| reflection over the x axis
h(x) = -|x| +5 add 5 to all the y values

Answer 10

This is basic translation of a function.

Answer 11

So this explains

y= -|x| + 5

but what if

y = |x-100| ? Will this shift the graph up/down or left/right?

Answer 12

It will shift it left

Answer 13

Yes! 100 units to the left.

So can you now sketch
f(x) = -|x-1|+4
?
Then ask when is it rising as you move from left to right

Answer 14

-3?

Answer 15

or 3

Answer 16

Take a step back and ask where the graph for
y= -|x| is rising

Answer 17

- infinity to infinity

Answer 18

Your graph will look like my sketch, just translated up 4 and to the left 1. Where would that be rising?

Answer 19

(1,3)

Answer 20

1,4

Answer 21

what does that mean that it rises from x = 1 to x=4?

Answer 22

from x=-3 to x=1

Answer 23

it the function not increasing from -5 to -3?