graphing exponential functions where exponent is a fraction

[prusso]

problem is to graph f(X)=3 ^.5x - 2 I can make a table of values of x and f(x) but every other f(x) has a square root of 3 in it - cant figure out how to graph those.....what am I missing? There are no examples like this in my book or on you tube.....

 

[wjm11]

problem is to graph f(X)=3 ^.5x - 2 I can make a table of values of x and f(x) but every other f(x) has a square root of 3 in it - cant figure out how to graph those.....what am I missing? There are no examples like this in my book or on you tube.....

Not sure I understand what your difficulty is. Why not just plug in even numbers for x and plot those (if you only want "nice" numbers to plot)?

 

[prusso]

Not sure I understand what your difficulty is. Why not just plug in even numbers for x and plot those (if you only want "nice" numbers to plot)?

With quadratic equations you can calculate vertex, figure out if parabola will go down or up, find x intercepts, find axisof symmetry all from the equation. With sin and cosine graphing there are similar things to help you graph - period, amplitude, etc. With exponential graphs in general you can find y asymptote, y intercept. So the question is are there similar elements to help graph when the exponent is a fraction - and do you expect growth if fraction of exponent is positive.....

 

[JeffM]

With quadratic equations you can calculate vertex, figure out if parabola will go down or up, find x intercepts, find axisof symmetry all from the equation. With sin and cosine graphing there are similar things to help you graph - period, amplitude, etc. With exponential graphs in general you can find y asymptote, y intercept. So the question is are there similar elements to help graph when the exponent is a fraction - and do you expect growth if fraction of exponent is positive.....

You simply ask questions about how the function behaves.

First, notice that \(\displaystyle f(x) = e^{0.5x} - 2\) is simply a shift downwards by 2 units of \(\displaystyle g(x) = e^{0.5x}\).

In short, f(x) = g(x) - 2. So if you can graph g(x), it is easy to graph f(x).

g(x) is everywhere positive and is asymptotic to the x-axis as x tends toward minus infinity. So f(x) will be asymptotic to x = - 2 as x tends to minus infinity.

g(0) = 1 so f(0) = 1 - 2 = - 1.

\(\displaystyle a > 0 \implies e^{0.5a} > 1 \implies g(x + a) = e^{(0.5x + 0.5a)} = e^{0.5x} + e^{0.5a} > g(x) + 1 > g(x).\)

So g(x) always increases as x increases, which means that f(x) always increases as x increases.

In my day, they used to teach this kind of curve sketching in calculus, where you have some extra tools to help.

If you need more than a sketch, calculate a few points with a calculator and draw a smooth curve joining them.

Is this what you were asking?

 

[HallsofIvy]

problem is to graph f(X)=3 ^.5x - 2 I can make a table of values of x and f(x) but every other f(x) has a square root of 3 in it - cant figure out how to graph those.....what am I missing? There are no examples like this in my book or on you tube.....

\(\displaystyle \sqrt{3}\) is approximately 1.7320508075688772935274463415059. What accuracy do you need to graph the points?