Secant lines: challenging problem 1

Consider the graph of the function f of x that passes through three points as shown. So these are the three points, and this curve in blue is f of x. Identify which of the statements are true. So they give us these statements. Let's see. This first one says f of negative a is less than 1 minus f of negative a over a. So this seems like some type of a bizarre statement. How are we able to figure out whether this is true from this right over here? So let's just go piece by piece and see if something starts to make sense. So f of negative a, where do we see that here? Well, this is f of negative a. This is the point x equals negative a. So this is negative a, and this is y is equal to f of negative a. So this is f of negative a right over here. And what we know about f of negative a, based on looking at this graph, is that f of negative a is between 0 and 1. So we can write that. 0 is less than f of negative a, which is less than 1. So that's all I can deduce about f of negative a right from the get-go. Now let's look at this crazy statement, 1 minus f of negative a over a. What is this? Well, let's think about what happens if we take the secant line, if we're trying to find the slope of the secant line, between this point and this point, if we wanted to find the average rate of change between the point negative a, f of negative a, and the point 0, 1. If this is our endpoint, our change in y is going to be 1 minus f of negative a. So 1 minus f of negative a is equal to our change in y. And our change in x, going from negative a to 0, so change in x is going to be equal to 0 minus negative a, which is equal to positive a. So this right over here is essentially our change in x over our change in y from this point to this point. It is our average rate of change from this point to this point. So it is our average rate of change, or you could say it's the slope of the secant line. So the secant line would look something like this. Slope of the secant line, so this right over here is slope of secant line between from f of negative a to 0 comma 1. So just looking at this diagram right over here, what do we know about this slope? And in particular, can we make any statements about that slope relative to, say, 0 or 1 or anything like that? Well, let's think about what a line of slope 1 would look like. Well, a line of slope 1, especially one that went through this point right over here, would look something like this. A line of slope 1 would look something like this. So this line right over here that I've just drawn that goes from negative 1 comma 0 to 0, 1, this has slope 1. So this slope is equal to 1. So if this green line has a slope of 1, does this blue line have a slope-- it clearly has a different slope. Is that slope, is this blue line steeper or less steep than the green line? Well, it's pretty clear that this secant line is steeper than the green line. It's increasing faster. So it's going to have a higher slope. So this, looking at it from this diagram, this blue line has a slope higher than 1. Or the slope of the secant line from negative a, f of negative a, to 0 comma 1, that is going to be greater than 1. So this thing right over here is greater than 1. So we're able to deduce, this thing right over here is less than 1. This thing over here is greater than 1. So this thing is less than that thing. So this must be true. Now let's look at this one. We're comparing the slope of the secant line. We're comparing the slope of the secant line that we just looked at. So this is the same value right over here. So we're comparing this slope right over here, to what's this? f of a minus 1 over a. Well, this is the slope of this secant line, this is the slope of this secant line that I'm drawing in this-- let me do it in more contrast. Let me do it in orange. That is the slope of this secant line. So which one has a higher slope? Well, it's pretty clear that the blue secant line has a higher slope than this orange secant line. But here, it's saying that the blue's slope is lower than the orange. So this is not going to be true. So this is not true. Then finally, let's look at this over here-- f of a minus f of negative a over 2a. So this is the slope. Let me draw this. So this right over here, this is the slope of the secant line between this point and this point right over here. Our change in y is f of a minus f of negative a. Our change in x is a minus negative a, which is 2a. So this is this secant line right over here. So let me draw it. So this secant line right over here, so they're comparing that slope to this slope. f of a minus 1 is our change in y, over a is our change in x. So we're comparing it to that one right over there. And you could immediately eyeball this kind of brownish, maroon-- I guess it's kind of a brown color-- this secant line that goes all the way from here to here is clearly steeper than this one right over here. And we know that, that the average rate of change from here to here is going to be higher than the average rate of change from here to here, because at least from negative a to 0, we were increasing at a much faster rate. And then we slowed down to this rate. So the average over the entire interval is definitely going to be more than what we get from 0 to a. So this one is also not true. This has a higher-- we actually know that this is false. Both of these would have been true if we swapped these signs around, if this was a greater than sign, if this was a greater than sign. So this is the only one that applies.