Interpreting a transformed polynomial function's graph
Narrator misspeaks '$11,005' (means $11,500) and '4,004', and at first says the shift is 'down by four' before correcting to $4,000 because the axis is in thousands. All are self-corrected; interventions use the corrected values.
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Bernard's profit over five months is modelled by p(x), measured in thousands of dollars. The video estimates the range of p(x) from its graph and shows that the transformed function d(x) = p(x) - 4 has the same shape shifted down 4 (thousand dollars), so both ends of its range are $4,000 lower.
You'll be able to
- explain what the range of a function means on a graph.
- estimate the range of a function from its graph over a given domain.
- describe how subtracting a constant from a function shifts its range, paying attention to units.
Key words
- range
- The set of all possible output (y) values a function takes.
- domain
- The set of input (x) values being used; here, 0 to 5 months.
- transformed function
- A new function made by changing an original one, such as d(x) = p(x) - 4.
- vertical shift
- Moving every point of a graph up or down by the same amount.
Transcript
- We're told that Bernard tracks the profit for his business for five months. The function p of x, and p of x is defined by all of this, models the amount of profit for Bernard's business of dollars, p of x over x months. The transformed function, d of x, which is p of x minus four, so essentially, you just get rid of this plus four up here, models a different profit scenario.
What is the best interpretation of the range of p of x as compared to d of x? So pause this video and think about it, and then we'll think about this together. All right, so let's just remind ourselves what range is. Range is the set of all of the possible outputs from a function, or if you think about it in this graph form, it's the range of all of the possible y values.
So let's write, let's make a little column here, range, and we have two functions here. We have our original function, p of x, and then we have our new transform function, d of x, which is p of x minus four. So first of all, p of x, what's the range of y values that it takes on over those five months?
So it looks like this minimum value right over here, I'm gonna eyeball it, but I'm gonna estimate it looks like it's about 3,500. Ideally, they would've told us exactly that, but it looks like, well, actually, it's not a profit of 3,500, it's a loss of 3,500. It's a -3,500 here.
And over the five months, the function would keep going, but we are bound, the domain, I guess we could say, is from zero to five months. So at five months, it looks like we are at, what is this? Looks like about halfway between 11 and 12? So this is $11,500. So the range looks like it gets as low as, including, so that's why I'm putting a bracket here, $3,500, 3,500.
Gee, let me not put a comma there. That might get confusing. $3,500, and it gets as high as $11,005. Oh, I said I wouldn't put commas there. $11,500. And now, let's think about d of x. d of x is essentially, at any given one, at the low point or the high point, everything is just shifted down by four.
So this right over here, this is 4,004. Yeah, this is listed in thousands of dollars, yes. So this is why, so thousands of dollars, so this is not just down by four, it's down by 4,000. So this is -7,500. So the low point is -7,500, if I'm estimating correctly, and the high point, it looks like it is...
It looks like it is $7,500, which is 4,000 less than 11,500, so 7,500. So the best interpretation of the range of p of x compared to d of x, you could say, well, the range gets shifted down by $4,000. The low point is 4,000 less, that's when the businesses had had a negative profit, and then the high point is $4,000 less as well.
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