Maths question

Hi, please could I have help on part c of this question? I dont understand why b has to be between 0 and 1?
Here is the question: https://app.gemoo.com/share/image-annotation/600044055366615040?codeId=DWaZnz5ekG7wr&origin=imageurlgenerator

Thank you!

What di dyou get for part (b)?

y = 4b^x is decreasing for x>0 so what does this imply for the value of b?

For part b i got (0,4). If the curve is negative, would it not mean that b is negative? Thanks!

Hi, please could I have help on part c of this question? I dont understand why b has to be between 0 and 1?
Here is the question: https://app.gemoo.com/share/image-annotation/600044055366615040?codeId=DWaZnz5ekG7wr&origin=imageurlgenerator

Thank you!

It's decreasing not negative ... when do powers of a number decrease?

If the value of b was negative, we would observe a different graph because even exponents will create a positive effect. We know that the graph is exponential and that the standard shape of the graph follows the same shape (maybe with some form of transformation) as graph A. However, the graph seems to be getting smaller with larger values of x, which suggests that the value of b has to be between 0 and 1.

Think about it. I will provide cases which proves this:

Why the value of b can't be negative?

Let's suppose that the value of b is -2. We would obtain the equation y = 4 x (-2)^x. When the power of x is even (let's take 2 for simplicity's sake but this is true for all even positive integers), we get 4 x (-2)^2, which is equal to 4x4 = 16. Therefore, we know that it can't be that because the graph crosses the y axis at (0,4) and the point at x = 2 is lower than that.

Why the value of b can't be greater than or equal to 1 or 0?

It can't be equal to 1 because we would just observe a straight line at y = 4 (and it's the same for b being equal to 0) and it can't be greater than 1 because we would observe normal exponential growth (as with graph A).

Why the value of b is then between 0 and 1?

When we have decimals raised to powers, as the power increases, the decimal gets smaller. For instance,

(1/2)^0 = 1
(1/2)^1 = 1/2
(0.5)^2 = 1/4
and this will keep on getting smaller and smaller and tend off to infinity (but we will not get onto that because it is not required for this question) and this is reflected by the behaviour of the graph as on the right hand side, as the value of x increases, the value of y decays (or decreases exponentially).

This shows that this works out for the powers going up from 0.

Now we have to look at the negative powers.

When we take a negative exponent for a given decimal, we essentially flip the fraction.

i.e if we have (0.5)^-1, we get 2/1 (which is the flipped version of 1/2) and that equates to 2.

Likewise, if we have (0.5)^-2, we get ((2)^2)/1, which is equal to 4.

So this shows us that as the power of x becomes more negative, the value of y gets larger and so this supports what we see on the graph therefore meaning that the value of b has to be between 0 and 1.

Now, most of this is deduction work. You don't have to actually show all of this working for a 2 mark question but it would be beneficial to show the substitution work perhaps?

What does the mark scheme say? Also, this is rather easy for an A level question 7. We do Edexcel and they are MUCH, MUCH HARDER (comparatively) but it's not a problem for me because I do Further Maths.

Hope this helps 😉 If you have any more Maths questions you require my assistance with, by all means post them on this forum and I will endeavour to get back to you on it. It will not only help you get to an answer but will also help me to consolidate my own knowledge so we both benefit from it.

Ah that makes sense, thank you very much!

If the value of b was negative, we would observe a different graph because even exponents will create a positive effect. We know that the graph is exponential and that the standard shape of the graph follows the same shape (maybe with some form of transformation) as graph A. However, the graph seems to be getting smaller with larger values of x, which suggests that the value of b has to be between 0 and 1.

Think about it. I will provide cases which proves this:

Why the value of b can't be negative?

Let's suppose that the value of b is -2. We would obtain the equation y = 4 x (-2)^x. When the power of x is even (let's take 2 for simplicity's sake but this is true for all even positive integers), we get 4 x (-2)^2, which is equal to 4x4 = 16. Therefore, we know that it can't be that because the graph crosses the y axis at (0,4) and the point at x = 2 is lower than that.

Why the value of b can't be greater than or equal to 1 or 0?

It can't be equal to 1 because we would just observe a straight line at y = 4 (and it's the same for b being equal to 0) and it can't be greater than 1 because we would observe normal exponential growth (as with graph A).

Why the value of b is then between 0 and 1?

When we have decimals raised to powers, as the power increases, the decimal gets smaller. For instance,

(1/2)^0 = 1
(1/2)^1 = 1/2
(0.5)^2 = 1/4
and this will keep on getting smaller and smaller and tend off to infinity (but we will not get onto that because it is not required for this question) and this is reflected by the behaviour of the graph as on the right hand side, as the value of x increases, the value of y decays (or decreases exponentially).

This shows that this works out for the powers going up from 0.

Now we have to look at the negative powers.

When we take a negative exponent for a given decimal, we essentially flip the fraction.

i.e if we have (0.5)^-1, we get 2/1 (which is the flipped version of 1/2) and that equates to 2.

Likewise, if we have (0.5)^-2, we get ((2)^2)/1, which is equal to 4.

So this shows us that as the power of x becomes more negative, the value of y gets larger and so this supports what we see on the graph therefore meaning that the value of b has to be between 0 and 1.

Now, most of this is deduction work. You don't have to actually show all of this working for a 2 mark question but it would be beneficial to show the substitution work perhaps?

What does the mark scheme say? Also, this is rather easy for an A level question 7. We do Edexcel and they are MUCH, MUCH HARDER (comparatively) but it's not a problem for me because I do Further Maths.

Hope this helps 😉 If you have any more Maths questions you require my assistance with, by all means post them on this forum and I will endeavour to get back to you on it. It will not only help you get to an answer but will also help me to consolidate my own knowledge so we both benefit from it.