1. **State the problem:** Determine which tables represent exponential functions and analyze given exponential functions for asymptotes, y-intercepts, domain, and range. 2. **Check each table for exponential behavior:** - Exponential functions have a constant ratio between consecutive y-values. 3. **Table a:** - Ratios: $\frac{-6}{-2} = 3$, $\frac{-18}{-6} = 3$, $\frac{-54}{-18} = 3$, $\frac{-162}{-54} = 3$ - Constant ratio 3, so table a is exponential. 4. **Table b:** - Ratios: $\frac{12}{4} = 3$, $\frac{24}{12} = 2$, $\frac{48}{24} = 2$, $\frac{144}{48} = 3$ - Ratios not constant, so not exponential. 5. **Table c:** - Differences: $18-12=6$, $24-18=6$, $30-24=6$, $36-30=6$ - Constant difference, so linear, not exponential. 6. **Table d:** - Ratios: $\frac{2}{1} = 2$, $\frac{4}{2} = 2$, $\frac{8}{4} = 2$, $\frac{16}{8} = 2$ - Constant ratio 2, so table d is exponential. 7. **Summary:** Tables a and d represent exponential functions. --- 8. **For each exponential function, find asymptote and y-intercept:** - General form: $y = a b^x + q$ - Horizontal asymptote: $y = q$ - y-intercept: value at $x=0$ is $y = a b^0 + q = a + q$ 9. a) $y = 2^x + 1$ - Asymptote: $y=1$ - y-intercept: $2^0 + 1 = 1 + 1 = 2$ 10. b) $y = 3(2)^x - 2$ - Asymptote: $y = -2$ - y-intercept: $3(1) - 2 = 1$ 11. c) $y = -2(2)^x + 3$ - Asymptote: $y=3$ - y-intercept: $-2(1) + 3 = 1$ 12. d) $f(x) = -4(2)^x + 5$ - Asymptote: $y=5$ - y-intercept: $-4(1) + 5 = 1$ 13. e) $y = 3^x - 1$ - Asymptote: $y=-1$ - y-intercept: $1 - 1 = 0$ 14. f) $y = -3^x - 1$ - Asymptote: $y=-1$ - y-intercept: $-1 - 1 = -2$ 15. g) $f(x) = 2(3)^x + 1$ - Asymptote: $y=1$ - y-intercept: $2(1) + 1 = 3$ 16. h) $y = 4(3)^x$ - Asymptote: $y=0$ - y-intercept: $4(1) = 4$ 17. i) $y = -2(4)^x + 3$ - Asymptote: $y=3$ - y-intercept: $-2(1) + 3 = 1$ 18. j) $f(x) = 5^x$ - Asymptote: $y=0$ - y-intercept: $1$ 19. k) $y = 0.5(2)^x + 1$ - Asymptote: $y=1$ - y-intercept: $0.5 + 1 = 1.5$ 20. l) $f(x) = -1.5(3)^x - 1$ - Asymptote: $y=-1$ - y-intercept: $-1.5 - 1 = -2.5$ --- 21. **Domain and range:** - Domain of exponential functions: all real numbers $(-\infty, \infty)$ 22. a) $y = 4^x$ - Range: $(0, \infty)$ because base > 1 and no vertical shift 23. b) $y = 2^x - 3$ - Range: $(-3, \infty)$ shifted down by 3 24. c) $y = 3(\frac{1}{4})^x + 4$ - Range: $(4, \infty)$ since $a=3>0$ and base $\frac{1}{4} < 1$, horizontal asymptote at $y=4$ 25. d) $y = -(\frac{1}{5})^x + 2$ - Range: $(-\infty, 2)$ because of negative leading coefficient and asymptote at $y=2$ --- 26. **Summary:** - Tables a and d are exponential. - Asymptotes and y-intercepts found for each function. - Domain always $(-\infty, \infty)$. - Range depends on $a$ and $q$. Final answers: - Exponential tables: a, d - Asymptotes and y-intercepts as above - Domains: all real numbers - Ranges as above