Now let's figure out theĮxponential function. To y is equal to five, so that is the y-intercept. Or another way you could've said it, if the slope is negative four, if this right over here is nine, you increase one in the x direction, you're gonna decreaseįour in the y direction, and that will get you Maybe this was 5.00001 or something, but now we know for sure Have guessed, actually, that the y-intercept here is five, but now we've solved it. Now, does that make sense that the y-intercept here is five? Well, you see that right over here. And so this part right over here, we could write that as negativeįour plus b is equal to one, and then we could add four toīoth sides of this equation, and then we get b is equal to five. So we could write f of one, which would be negativeįour times one plus b. Use either one of these points to figure out, given an x, what f of x is, and then we can solve for by. You are decreasing your y by four there, so that makes sense that So every time you increase your x by one, you are decreasing your y. Every time you increase your x by one, you're decreasing your y. And so now we can write that f of x is equal to negative four, negative four, that's our slope, times x. We get negative eight over two, which is equal to negative four. Negative one, y equals nine, and so we just took the differences. Another way to think about it, the way I drew it right over here, we're finishing at xĮquals one, y equals one. Let me do this in maybeĪnother color here. And what about our change in y? Well, we start at nine. So we could think of itĪs we're finishing at one. So let's see, between those two points, what is our change in x? Our change in x, we're going from x equals The rate and change of the vertical axis with respect to the horizontal axis. We could use those pointsįirst to figure out the slope. But let's use the data they're giving us, the two points of intersection, to figure out what the equations The fact that g of x keeps approaching, it's getting closer andĬloser and closer to zero as x increases. Our r right over here, they tell us that r is greater than zero, but it's a pretty good hint that r is going to beīetween zero and one. Larger and larger and larger is a pretty good hint that This exponential function keeps decreasing as x gets The exponential function, so right over here.
Negative one comma nine, so this is negative oneĬomma nine right over here, and one comma one.
The linear function f of x is equal to mx plus b and the exponential function g of x is equal to a times r to the x where r is greater than zero pass through the points
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