Need help with modulus functions and how to present answers

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Polynomial112233

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As per attachment, question 10.

A=1/2
B=—6
C=2

I got these answers but have difficulties in writing them in proper presentation.

Pretty lost for part ii where they mention "state", because I cannot think of an easy way to get the ranges of m.

Thanks in advanced for the help!
 

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What do we know about the line? Maybe some points it passes through?
 
I understand. I'm asking whether you can make some conclusions based on the equation of the line.
 
But the line is defined as y = mx + 4. You do know a point on the line: its y-intercept! I assume that's what lev was getting at.

The easy way to solve this is to plot that point on the graph, and just think about what lines through that point will intersect the graph in only one point.

You could instead solve the equation c - |ax + b| = mx + 4 (with a, b, c filled in) and determine under what condition there is only one solution. That will be harder, but perhaps more convincing.
 
Ah, I get that what lev is talking about now. Knowing that is passes through (0,4), how can we prove that it passes through only one point?

For part ii, I was wondering if there is a more straightforward way to looking at the range of m because they simply mention *state*. With x being an unknown and requiring the the range of m, I am pretty lost since we have 2 unknowns and one equation
 
lev888 said:
Do you know how changing m affects the line?
Click to expand...
I do. It affects the gradient.

Conceptually the only way I can think about presenting this is to either touch the "turning point" , or to have the m such that it parallel or divergent to the right side of the line.

Please do correct me if my concept is wrong. And if it is right, I'm also thinking about is how to present it in an examination format so that it is easy to understand for the marker
 
Not sure what you mean by parallel or divergent to the right side of the line. Would this result in intersection in one point?
But touching the max value point is a good approach. Knowing two points the line passes through should help you find m.
 
Polynomial112233 said:
I do. It affects the gradient.

Conceptually the only way I can think about presenting this is to either touch the "turning point" , or to have the m such that it parallel or divergent to the right side of the line.

Please do correct me if my concept is wrong. And if it is right, I'm also thinking about is how to present it in an examination format so that it is easy to understand for the marker
Click to expand...
I think you have the right idea.

Try writing a rough draft of an answer, starting with a description of what you are doing, and ending with a statement about what values of m will make the line intersect at one point. Then we can discuss how to improve the format.
 
This is what I have thus far.

Answer key mentions m > 1/2 which I do not understand why. I'm thinking it's a mistake but I hope someone can correct me.
 

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I would love to see alternate ways of solving by equation if any, always love to see different methods of solving. Thanks!
 
I don't have time to examine all the details right now (or maybe for the next day or two). But it would be very helpful, if you ask what they mean in the answer key, for you to show us what the answer key says! I would expect it to mention that when m>1/3 and m<-1/3, the line will intersect once (if I'm not thinking too fast). Have you considered positive slopes? Have you noticed what your third picture tells you? (The maximum point is not the only place where there can be a single intersection.)
 
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