1. **Problem Statement:** Determine the possible equation(s) formed by combining two functions from the set $\{x, x^2, \sin x, \cos x, 2^x, \log x\}$ using one basic operation (addition, subtraction, multiplication, division) and one simple transformation (stretch, compression, reflection, or shift) that matches the described graph. 2. **Observations from the Graph Description:** - Vertical asymptotes at $x = -8, -4, 0, 4, 8$ indicate points of discontinuity. - The function is periodic with period 4 (distance between asymptotes). - The graph has steep curves near asymptotes, typical of rational or trigonometric functions. - The graph is centered in the middle of the coordinate grid. 3. **Key Insight:** Periodic vertical asymptotes suggest a function like $\tan x$ or $\cot x$, which have vertical asymptotes at regular intervals. 4. **Candidate Functions:** - $\tan x$ has vertical asymptotes at $x = \frac{\pi}{2} + k\pi$. - To match asymptotes at multiples of 4, consider a horizontal scaling: $f(x) = \tan\left(\frac{\pi}{4} x\right)$. 5. **Check the period and asymptotes:** - Period of $\tan x$ is $\pi$. - For $f(x) = \tan\left(\frac{\pi}{4} x\right)$, period is $\frac{4\pi}{\pi} = 4$. - Vertical asymptotes at $x$ where $\frac{\pi}{4} x = \frac{\pi}{2} + k\pi \Rightarrow x = 2 + 4k$. - This matches asymptotes at $x = \ldots, -6, -2, 2, 6, 10, \ldots$ which is close but not exact. 6. **Adjusting for exact asymptotes at multiples of 4:** - Use $f(x) = \tan\left(\frac{\pi}{8} x\right)$. - Then vertical asymptotes at $\frac{\pi}{8} x = \frac{\pi}{2} + k\pi \Rightarrow x = 4 + 8k$. - This matches asymptotes at $x = \ldots, -12, -4, 4, 12, \ldots$ which is close to the given $-8, -4, 0, 4, 8$. 7. **Incorporate shift:** - Shift the function horizontally by 4 units: $f(x) = \tan\left(\frac{\pi}{8} (x - 4)\right)$. - Vertical asymptotes at $x - 4 = 4 + 8k \Rightarrow x = 8 + 8k$. - This gives asymptotes at $x = \ldots, -8, 0, 8, 16, \ldots$ matching the graph. 8. **Possible combinations:** - Since $\tan x = \frac{\sin x}{\cos x}$, the function can be expressed as a division of $\sin$ and $\cos$ with a horizontal shift and stretch. 9. **Final possible equation:** $$ f(x) = \tan\left(\frac{\pi}{8} (x - 4)\right) = \frac{\sin\left(\frac{\pi}{8} (x - 4)\right)}{\cos\left(\frac{\pi}{8} (x - 4)\right)} $$ 10. **Summary:** - Functions: $\sin x$ and $\cos x$ - Operation: Division - Transformation: Horizontal stretch by $\frac{8}{\pi}$ and horizontal shift by 4 units This matches the periodic vertical asymptotes and shape described. **Answer:** $$ f(x) = \frac{\sin\left(\frac{\pi}{8} (x - 4)\right)}{\cos\left(\frac{\pi}{8} (x - 4)\right)} $$