I can't find an appropriate piecewise function for this graph

Question

On one of my piecewise questions I've split a graph into an exponential function, a cosine function and a parabolic function. I've done fine for exponential and parabola but I'm totally stuck on cosine and I have no idea what I'm doing... Please help!

Answer 1

One issue is that the curve you have sketched has a continuous derivative, which rounds things off at the joints. The part you have labeled exp fits $y=2^x$ perfectly, but that is growing rapidly at $x=2$ and doesn't blend with the cos piece. If we let there be corners at $x=2,5$ it is not so hard to find a piecewise graph. $$y=\begin {cases} 2^x&-2 \le x \le 2\\\frac 52 +\frac 32 \cos \frac {x-2}\pi x& 2\lt x \le 5\\2+(x-6)^2&5\lt x \le 7 \end {cases}$$ If you want things to blend smoothly, you need more freedom in the function.

Answer 2

I assume you want to find the equation of a cosine-shaped curve which has been possibly scaled and translated. This means that the curve has the form:

$$y(x) = a\cdot \cos{(bx + c)} + d$$

And all you have to do is identify the missing numbers $a, b, c, d$.

Here are some ways to analyze your cosine-shaped curve to find out how much it has been scaled and translated.

• First, if you know the maximum and minimum height of the curve, then their average will be the value of $d$. (Every cosine-shaped curve oscillates around its midline, the horizontal line $y=d$.)
• Second, if you know the maximum and minimum height of the curve, their difference is the overall height of your curve. Ordinarily, the curve $y(x)=\cos{x}$ spans from -1 to 1 and consequently has height 2. If the curve instead has been stretched vertically so that it has height $v$, you know that $a = v/2$.
• Third, if you know the period of the curve (i.e. how long it takes for it to repeat itself), then you can compute $b$. Ordinarily, the curve $y(x) =\cos{x}$ repeats every $2\pi$ units. If it has been stretched horizontally so that it repeats every $h$ units instead, then you know that $b = 2\pi / h$.
• Finally, how far has the curve been translated horizontally? There are many good answers. If you can find the x-coordinate $x$ of any point where the curve reaches its maximum height, then you can use $c\equiv -x$.