Larger, a little bit larger, but you'll see that we Have 3 to the 0 power, which is just equal to 1. And then 3 to the negativeĢ power is going to be 1/9, right? 1 over 3 squared, and then we Super-small number to a less super-small number. To the third power, which is equal to 1/27. When x is equal to negativeĭifferent color. ![]() The negative 4 power, which is equal to 1 over 3 to Y-values are going to be for each of these x-values. See how quickly this thing grows, and maybe we'llĮqual to negative 4. The third power, this is 3 to the x power. So let's just write an exampleĮxponential function here. ![]() Really just show you how fast these things can grow. Introduce you to the idea of an exponential function and This shifts from the origin to (-2,-5) which makes the asymptote at y=-5, but it is a little harder to determine the x axis shift back 2. This is a good introduction, which is good for all but shifts in the x direction such as y = 3 (4)^(x+2) - 5. Thus, you would have to do (5- 3)/(4 - 3) to get 2/1=2 as the base. Thus y=2^x + 3 would have points (0,4) 1 away from asymptote, (1,5) two away from asymptote, etc. If you see an asymptote at say y=3, then "act like" this is the y axis and see how far points are away from the this line. This can be easily be determined by a change in the asymptote. So the next easiest is to shift up and down by adding a constant to the end. This will work the same for decay functions, but the base will be a fraction less than 1. Similarly, if we have (0,3) and (1,6) our base is 6/3=2, but the scale factor is 3, so we have y=3(2)^x. ![]() With 2(2)^x, you double all the y values to (0,2)(1,4)(2,8)(3,16) - note that 16/8=8/4=4/2=2, so we still get the same base, but the y intercept tells us that the scale factor is 2. Next, if we have to deal with a scale factor a, the y intercept will tell us that. It works the same for decay with points (-3,8). Thus, we find the base b by dividing the y value of any point by the y value of the point that is 1 less in the x direction which shows an exponential growth. So we find the common ratio by dividing adjacent terms 8/4=4/2=2/1=2. You start with no shifts in x or y, so the parent funtion y=2^x has a asymptote at y=0, it goes through the points (0,1) (1,2)(2,4)(3,8). Learning the behavior of the parent functions help determine the how to read the graphs of related functions. So one basic parent function is y=2^x (a=1 and b=2). ![]() So the standard form for a quadratic is y=a(b)^x.
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