Oturum Detayları - 4139cf86-c895-4491-a5e4-0ffd8e5475a0

06:52:24

Öğrenci

Yes, I am ready! Can you explain this topic in detail?

06:52:25

💭 AI Öğretmen

Getting the details ready

06:52:25

Görsel

06:52:27

AI Öğretmen

Graphs in mathematics are always read from left to right, just like reading a sentence in English.

06:52:29

AI Öğretmen

This idea is important because it helps us understand what a function is doing as we look along the graph.

06:52:30

AI Öğretmen

When we have two $x$-values, we compare them by moving from the left one to the right one.

06:52:31

AI Öğretmen

We look at how the function changes as we move from the first $x$-value to the second.

06:52:32

AI Öğretmen

If the function goes up, it is increasing, and if it goes down, it is decreasing.

06:52:34

AI Öğretmen

Remembering to read the graph from left to right helps us find and describe the intervals where the function is increasing or decreasing.

06:52:36

AI Öğretmen

Did you understand how reading graphs from left to right helps identify increasing and decreasing intervals?

06:53:08

Öğrenci

I got it

06:53:11

💭 AI Öğretmen

Excellent, moving forward

06:53:11

🎥 Video

06:53:12

AI Öğretmen

Did you understand how to identify an increasing interval on a graph?

06:53:54

Öğrenci

I got it

06:53:56

💭 AI Öğretmen

Great, preparing a question

06:53:56

❓ Soru basic

![Figure](https://pajxpzsolirfrzhpksii.supabase.co/storage/v1/object/public/question-images/question_33e16c0c-dc3a-47bd-91c9-e1d47eecd95c.png) Example: The graph of f is shown on the below. Find the intervals of x for which f(x) is **increasing**.

06:55:48

Öğrenci

Can you give me a hint, teacher?

06:55:49

💭 AI Öğretmen

Let me consider that

06:55:51

AI Öğretmen

What does it mean for a function to be increasing when you look at the graph from left to right?

06:55:53

AI Öğretmen

Can you identify where on the graph the function values are going uphill as you move from left to right?

06:55:54

AI Öğretmen

How can you use the x-values at the start and end of these "uphill" sections to write the intervals where the function is increasing?

06:56:05

Öğrenci

My answer is: -2,2u 3,3

06:56:05

💭 AI Öğretmen

Reviewing your answer now

06:56:12

AI Öğretmen

Your answer was not correct, let me explain why. The intervals where the function is increasing are the parts of the graph where the curve goes upward as you move from left to right.

06:56:14

AI Öğretmen

Looking at the graph, from $x = -5$ to $x = -2$, the function rises from $y = 0$ to $y = 2$, so it is increasing there. Then, after a dip, it increases again from $x = 0$ to $x = 3$.

06:56:16

AI Öğretmen

It seems you wrote points or numbers like "-2,2u 3,3" which don't correctly show intervals or the correct ranges where the function rises. Remember, intervals are written with brackets like $[-5, -2]$ and $[0, 3]$.

06:56:18

AI Öğretmen

For example, if a function goes up from $x=1$ to $x=4$, we say it is increasing on $[1, 4]$. So here, the correct increasing intervals are $[-5, -2]$ and $[0, 3]$.

06:56:20

AI Öğretmen

Therefore, the correct answer should be $[-5, -2] \cup [0, 3]$.

06:56:20

AI Öğretmen

Should we continue with the lecture?

06:57:19

Öğrenci

Alright, let's continue

06:57:20

💭 AI Öğretmen

Proceeding to next part

06:57:30

Görsel

06:57:34

AI Öğretmen

A function is called decreasing on an interval if, whenever you pick two numbers $x_1$ and $x_2$ in that interval where $x_1$ is less than $x_2$, the value of the function at $x_1$ is greater than the value at $x_2$.

06:57:37

AI Öğretmen

Think of it as walking down a hill: as you move forward along the path, your height gets lower and lower.

06:57:39

AI Öğretmen

Another example is your phone’s battery percentage during the day; as time goes by, the percentage usually gets smaller, showing a decreasing function.

06:57:40

AI Öğretmen

Did you understand how a decreasing function works with the examples given?

06:58:07

Öğrenci

I got it

06:58:10

💭 AI Öğretmen

Let us review this next step

06:58:10

❓ Soru basic

![Figure](https://pajxpzsolirfrzhpksii.supabase.co/storage/v1/object/public/question-images/question_fb69fa74-e294-4d91-a021-e549025bd3c4.png) Example: The graph of $f$ is shown on the above. Find the intervals of $x$ for which $f(x)$ is **decreasing**.

06:59:40

Öğrenci

My answer is: f(3)

06:59:41

💭 AI Öğretmen

Let me review your answer

06:59:46

AI Öğretmen

Your answer was not correct, let me explain why. The question asks for the interval where the function is decreasing, which means where the graph goes down as you move from left to right, not just a single point like f(3).

06:59:47

AI Öğretmen

A function is decreasing on an interval if for every two points in that interval, the function value at the first point is greater than the function value at the second point. This means the graph slopes downward in that interval.

06:59:50

AI Öğretmen

Looking at the graph, the function decreases between x = -2 and x = 0 because the curve goes down from the peak at (-2, 2) to (0, 0). Before -2 and after 0, the function either increases or stays constant.

06:59:53

AI Öğretmen

A common mistake might be confusing a point on the graph with an interval or misunderstanding what "decreasing" means. Remember, it's about the direction of the graph over a range, not just a value at a point.

06:59:55

AI Öğretmen

For example, if a function goes from y=5 at x=1 down to y=2 at x=3, it is decreasing on the interval [1,3]. But if it goes from y=2 at x=3 up to y=4 at x=5, it is increasing on [3,5].

06:59:57

AI Öğretmen

Therefore, the correct answer should be the interval [-2, 0], where the function is decreasing.

06:59:57

AI Öğretmen

Should we continue with the lecture?

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