Transformation of a trig function

functionstransformationtrigonometry

From a fill-in-the-blank type of question:

The graph of $y=\cos(15x+13)$ is obtained by $\underline{\text{shrinking}}$ the graph of $y=\cos(x)$ in the $\underline{ x}$ direction by a factor of $\underline{15}$ and then shifting $\underline{(?)}$ units to the $\underline{\text{left}}$.

For the shift of the function, I know that the function will shift $13$ units to the left, but the answers that I can choose from don't have $13$ as an answer. The answers I can choose from are:

  • $\frac{-2}{15}$
  • $\frac{43}{15}$
  • $\frac{-17}{15}$
  • $\frac{28}{15}$
  • $\frac{13}{15}$

What am I missing here?

If $y=f(x+c)$ shifts the graph of the function $c$ units to the left, then why is $13$ not an answer? Is it that I'm missing something?

Best Answer

Bear in mind that, if you want to stretch/translate $f(x)$, then the function $g(x)$ given by

$$g(x) = a f(b(x-c)) + d$$

has the graph of $f$ but

  • stretched by a factor of $|a|$ vertically (with a flip for $a<0$)
  • stretched by a factor of $|b|$ horizontally (with a flip for $b<0$)
  • shifted right by $c$ units
  • shifted up by $d$ units

(You can play with this in a Desmos demo here.)

Of particular note is that we have $f(b(x-c))$, not $f(bx-c)$.

Hence to get the appropriate shift in your case, you need to bring it into this form, by factoring the coefficient of $x$. This gives you

$$\cos(15x + 13) = \cos \left( 15 \left( x + \frac{13}{15} \right) \right)$$

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