I have a question regarding the plot of a exponent function. Basically, how do you construct the graph? Let's say we have: y=2^x
Should I just make a table, insert some x values, let's say x|-3|-2|-1|1|2|3. Then I proceed replacing the x with my given values and calculate y.
Afterwards I insert the points into the axis.
Is that the correct method, or am I doing something wrong? I also have a to make a plot for a function like this: y=e^x, what exactly is e? I read that e has a specific value, much like pi, but then again, I'm not sure.
Much appreciated if anybody can help!
Should I just make a table, insert some x values, let's say x|-3|-2|-1|1|2|3. Then I proceed replacing the x with my given values and calculate y.
Afterwards I insert the points into the axis.
Is that the correct method, or am I doing something wrong? I also have a to make a plot for a function like this: y=e^x, what exactly is e? I read that e has a specific value, much like pi, but then again, I'm not sure.
Much appreciated if anybody can help!
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You definitely seem to be thinking along the right lines.
When you're starting to graph exponentials like y = 2^x, you really do have to start with a table. Find the value of y for various values of x. Plot the points, and try to connect them. Some key points:
1) The graph "lives" above the x-axis, and never touches it. That is because 2^x is ALWAYS positive. 2^-999 is a very small, but positive, number....like 0.00000000000000000000001.
2) On the far left side of the graph, where x = -1000 and x=-999, it looks fairly flat. That's because there's not much ABSOLUTE difference between 2^-1000 and 2^-999, unless you really, really, really zoom in. Absolute difference would be 2^-1000 - 2^-999, and that's a tiny number. However, there is a signficiant RELATIVE difference...2^-999 is TWICE as much as 2^-1000. However, it's twice a very small number.
3) The line y=0 never touches the graph, but on the left side, it "approaches" y =0. y=0 is called an asymptote. You may have seen asymptotes before, particularly in y = 1/x, where both y=0 and x=0 are asymptotes.
4) On the right side of the graph, it SHOOTS up really, really fast. Exponential growth is POWERFUL. 2^10 = 1024, but 2^11 = 2048....which is a HUGE absolute difference! You'll hear later on that exponential growth is "faster" than polynomial growth. That is to say, 2^x will, at some point, totally overpower x^2....it will even eventually overpower x^1000.
When you're starting to graph exponentials like y = 2^x, you really do have to start with a table. Find the value of y for various values of x. Plot the points, and try to connect them. Some key points:
1) The graph "lives" above the x-axis, and never touches it. That is because 2^x is ALWAYS positive. 2^-999 is a very small, but positive, number....like 0.00000000000000000000001.
2) On the far left side of the graph, where x = -1000 and x=-999, it looks fairly flat. That's because there's not much ABSOLUTE difference between 2^-1000 and 2^-999, unless you really, really, really zoom in. Absolute difference would be 2^-1000 - 2^-999, and that's a tiny number. However, there is a signficiant RELATIVE difference...2^-999 is TWICE as much as 2^-1000. However, it's twice a very small number.
3) The line y=0 never touches the graph, but on the left side, it "approaches" y =0. y=0 is called an asymptote. You may have seen asymptotes before, particularly in y = 1/x, where both y=0 and x=0 are asymptotes.
4) On the right side of the graph, it SHOOTS up really, really fast. Exponential growth is POWERFUL. 2^10 = 1024, but 2^11 = 2048....which is a HUGE absolute difference! You'll hear later on that exponential growth is "faster" than polynomial growth. That is to say, 2^x will, at some point, totally overpower x^2....it will even eventually overpower x^1000.
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