00:02
Okay, so for this question, we're basically going to go through and find whether the given the given coordinate points are linear or exponential functions.
00:12
And then if they are linear or exponential, then we're going to provide the function for it.
00:19
So to start with, we can look over here.
00:21
These are the general forms for the linear and exponential functions.
00:25
So for linear you have y equals a x plus b, where x is your variable.
00:32
A and b are your two constants.
00:35
And the same thing for your exponential form, which is y equals b times a to the power of x.
00:41
Or again, x is a variable, and b and a are the two constants.
00:45
So to start with, let's do this one.
00:50
So basically, an exponential function, if this is an exponential function, then the ratios of every single f of x or y point for its corresponding x point in increment.
01:02
Of one, every consecutive one compared to the previous one will have the same constant ratio.
01:10
So what that means is, for example, if we take this right here, this one and this one, and then we compare them like so.
01:18
So we take the ratio between the second and the first one, so the first two, we find that the ratio is actually two.
01:26
So we do that with the next one, 12 and 6, and we also, again, we get two.
01:34
But if we do that with this one, 18 and 12, then we actually, 18 over 12 is equal to 1 .5.
01:41
So the ratio is not right.
01:44
So since it's not constant, this is not an exponential function.
01:49
And next to check if it's a linear function, a linear function basically for every increment of x by 1, f of x changes by the same amount.
01:57
So the difference, in this case, instead of the ratio, the difference between each consecutive point has to be constant.
02:04
So the way we check that is just by going through and finding the difference.
02:08
So the first one, the first two points, the difference is three, but the next two points, the difference is actually six.
02:16
So what that means is this is not an exponential function or a linear function.
02:23
So this is actually neither exponential or linear.
02:28
So that's the answer for this first one.
02:31
So now if we go on to the next one, we can once again do the same thing.
02:37
But instead of simply starting with just checking exponential and then checking linear, we can actually just look at it and see that the points here, 25, 8, 11, 14 are there basically you can tell that they're linearly changing by a constant amount.
02:57
So if we check over here and we go back to 5 .5.
03:01
Minus 2 we can see that that's 3 8 minus 5 that's also 3 and then we go 11 minus 8 and that's also 3 and 14 minus 11 which is also equal to 3 so since the differences are constant then that basically means that this is a linear function so this is a linear function and then we can find the we can find basically the the the the the function for it from basically just looking at the general form of the function, which we found was y equals a x plus b.
03:47
And in this case, a would be 3 because that's how much it's changing by.
03:52
So you'd get g of x equals 3 plus and b, b is when x is equal to 0.
04:02
So whenever x is equal to 0, the corresponding y value or g of x value, is the value of b.
04:09
So in this case, it's actually five.
04:10
So your thing would be, your function would be, g of x equals 3x plus 5.
04:15
Now if you go back and you plug stuff in, so for example, if we plug in 2 here, sorry, if we plug in x here for negative 1, then we would get negative 3 plus 5, which would be 2.
04:26
So that this is your final answer for this one.
04:33
Now i was going to next one.
04:39
And here the process is going to be really pretty much the exact same.
04:43
So let's check linear first because that's a simpler one.
04:48
I'll switch to black here.
04:49
So we'll check linear first.
04:50
You'd get 1 minus 1 over 4, and that would be 3 over 4.
04:55
And we check the next one, 4 minus 1.
04:58
Well 4 minus 1 gives you 3.
04:59
So these two aren't the same, so it's not linear.
05:03
So for exponential, once again, we'll check the ratios.
05:07
So 1 over 1 over 4.
05:09
That gives you basically 4.
05:12
Then you get 4 over 1, which also gives you 4.
05:14
16 over 4, which also gives you 4, then 64 over 16, which also gives you 4.
05:27
So the ratios here are constant.
05:30
So that means this is indeed an exponential function.
05:36
And we can find the general form, or sorry, the function from the general form, which is, as we saw earlier, y equals b times a to the power of x.
05:47
In this case, a is 4, of course, because that is the constant ratio.
05:56
And we just need to find b, which is once again the exact same.
05:59
Whenever x is equal to zero, the corresponding y value would be b.
06:03
In this case, it's simply one.
06:05
So we don't need to indicate anything there.
06:07
So we'd simply get h of x is equal to 4 to the power of x.
06:12
And that would be your final answer for this one.
06:18
So now let's move on.
06:23
Actually, i'm just circle this one.
06:29
Now for this one.
06:32
So once again, we'd be doing pretty much the same thing.
06:37
Just by looking at it, you can sort of tell that the linear, the linearity of this is not going to be constant because it's going to be two -thirds to one, which would be an increase of one -third...