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nycguitarguy

Member

Registered: 2024-02-24

Posts: 542

Function Decreasing Over Interval

Suppose that the function y = f(x) is decreasing on the interval (-2, 7).

A. What can be said about the graph of y = -f(x)?

B. What can be said about the graph of y = f(-x)?

nycguitarguy

Member

Registered: 2024-02-24

Posts: 542

Re: Function Decreasing Over Interval

Let me see.

For A

We can say that the graph of y = -f(x) has reflection about the x-axis. Multiply the function by -1.

For B

We can say that the graph of y = f(-x) has reflection about the y-axis. Replace every x in the function by -x.

Is any of this correct?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,196

Re: Function Decreasing Over Interval

They want you to say if the transformed graph has a decreasing or increasing section and if so where.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

nycguitarguy

Member

Registered: 2024-02-24

Posts: 542

Re: Function Decreasing Over Interval

They want you to say if the transformed graph has a decreasing or increasing section and if so where.

Bob

Oh, I see. I will try again. Thank you for your input.

nycguitarguy

Member

Registered: 2024-02-24

Posts: 542

Re: Function Decreasing Over Interval

They want you to say if the transformed graph has a decreasing or increasing section and if so where.

Bob

I haven't been able to figure out these two. Can you help me?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,196

Re: Function Decreasing Over Interval

You did the hard bit in post 2. 

A graph that is decreasing means it slopes downwards as you go from left to right x=2 to x=7.

If you reflect it what does that do to  the slope?

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

nycguitarguy

Member

Registered: 2024-02-24

Posts: 542

Re: Function Decreasing Over Interval

You did the hard bit in post 2. 

A graph that is decreasing means it slopes downwards as you go from left to right x=2 to x=7.

If you reflect it what does that do to  the slope?

Bob

Let me see.

Let m = slope

I know that x = 2 means y = 0 and x = 7 means y = 0.

m = (0 - 0)/(7 - 2)

m = 0/5

m = 0

Wait...

This makes no sense. Let me get back to you later.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,196

Re: Function Decreasing Over Interval

I have made up a function, shown in red, that is decreasing from x = -2 to x = +7

-f(x) is a reflection in the x axis. I have shown that graph in blue.

From x = -2 to x = + 7 the blue graph is increasing.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

nycguitarguy

Member

Registered: 2024-02-24

Posts: 542

Re: Function Decreasing Over Interval

They want you to say if the transformed graph has a decreasing or increasing section and if so where.

Bob

Part A

The graph of y = -f(x) reflect the graph of y = f(x) about the x-axis.
If f(x) is decreasing on an interval, then -f(x) will be increasing on the same interval.

You say?

Part B

The graph of y = f(-x) reflects the graph of y = f(x) about the y-axis.
If f(x) is decreasing on an interval, then f(-x) will also be decreasing on that same interval.

You say?

nycguitarguy

Member

Registered: 2024-02-24

Posts: 542

Re: Function Decreasing Over Interval

I have made up a function, shown in red, that is decreasing from x = -2 to x = +7

-f(x) is a reflection in the x axis. I have shown that graph in blue.

https://i.imgur.com/DvjpEby.gif

From x = -2 to x = + 7 the blue graph is increasing.

Bob

When the red graph is decreasing, the blue graph is increasing.

When the blue graph is increasing, the red graph is decreasing.

The graph of y = -f(x) is decreasing at the point (7, 0).

The graph of y = f(x) is increasing at the point (7, 0).

You say?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,196

Re: Function Decreasing Over Interval

When the red graph is decreasing, the blue graph is increasing.

When the blue graph is increasing, the red graph is decreasing.

Correct.

The graph of y = -f(x) is decreasing at the point (7, 0).

The graph of y = f(x) is increasing at the point (7, 0).

The graph I made up had zero gradient at the end points of the interval.

The original question describes the interval as (-2, 7) . Don't get confused here with coordinates. This doesn't mean x=-2, y = 7; but rather all x values from -2 to +7. As the brackets are round not [ ] the end points are not included so it's ok for me to make up a graph that stops decreasing at the endpoints.  If it's not decreasing it must either have an instantaneous change to positive or go through zero gradient at a local maximum or minimum. I chose the latter as it's easier to make up an equation for this.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

nycguitarguy

Member

Registered: 2024-02-24

Posts: 542

Re: Function Decreasing Over Interval

When the red graph is decreasing, the blue graph is increasing.

When the blue graph is increasing, the red graph is decreasing.

Correct.

The graph of y = -f(x) is decreasing at the point (7, 0).

The graph of y = f(x) is increasing at the point (7, 0).

The graph I made up had zero gradient at the end points of the interval.

The original question describes the interval as (-2, 7) . Don't get confused here with coordinates. This doesn't mean x=-2, y = 7; but rather all x values from -2 to +7. As the brackets are round not [ ] the end points are not included so it's ok for me to make up a graph that stops decreasing at the endpoints.  If it's not decreasing it must either have an instantaneous change to positive or go through zero gradient at a local maximum or minimum. I chose the latter as it's easier to make up an equation for this.

Bob

Do we need to find the increasing or decreasing interval for both A and B?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,196

Re: Function Decreasing Over Interval

I think so.

It would have been better if the questioner had said "For each function identify if it is decreasing, increasing or neither including where in the domain this occurs." But they possibly thought it was obvious.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

nycguitarguy

Member

Registered: 2024-02-24

Posts: 542

Re: Function Decreasing Over Interval

I think so.

It would have been better if the questioner had said "For each function identify if it is decreasing, increasing or neither including where in the domain this occurs." But they possibly thought it was obvious.

Bob

Can you finish this one by providing the interval for both A and B?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,196

Re: Function Decreasing Over Interval

Both new functions reflect the original in an axis. So it's the same for both: increasing from -2 to + 7.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

nycguitarguy

Member

Registered: 2024-02-24

Posts: 542

Re: Function Decreasing Over Interval

Both new functions reflect the original in an axis. So it's the same for both: increasing from -2 to + 7.

Bob

Beautiful. We can now move on.