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The problem wanted me to find the negative energy solutions to the Schrodinger equation for a periodic potential. I did that and graphed it here:

enter image description here

However, my graph isn't turning out like the given answer's and I want to understand why. This is what is given:

enter image description here

The equation 5.66 that the author mentions is $f(z) = \cos(z)+\beta \frac{\sin(z)}{z}$.

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    I think the function you plotted on WolframAlpha is only supposed to apply for $z<0$. – 2017-01-17

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I think what Ian said is right (that your function should only be plotted on negative z). Also on your wolfram alpha plot what you meant to do was make the range -1 to 1 but instead you have just plotted -1 and 1 on top of your function while the range got too big so wolfram alpha gave up.