Graph Exponential, Logarithmic, and Trig Functions - Algebra 2

1. What transformation does $f(x)=b^{(x-h)}$ represent relative to $y=b^x$?

   Horizontal shift right by $h$ (left if $h<0$). Input transformation: subtracting shifts graph right.

2. State the five key $x$-values for one cycle of $y=\sin(x)$ from $0$ to $2\pi$.

   $0,\,\frac{\pi}{2},\,\pi,\,\frac{3\pi}{2},\,2\pi$. Standard quarter-period intervals for sine function.

3. What is the midline of $y=-3\sin(2x)+4$?

   $y=4$. Vertical shift parameter is $k=4$.

4. What is the amplitude of $y=\frac{1}{2}\cos(x)-1$?

   $\frac{1}{2}$. Coefficient magnitude is $|\frac{1}{2}|=\frac{1}{2}$.

5. What is the general form of an exponential function used for graphing transformations?

   $f(x)=a\,b^{(x-h)}+k$ with $b>0$ and $b\neq 1$. Standard form showing all transformations with base restrictions.

6. What is the period of $y=\sin(3x)$?

   $\frac{2\pi}{3}$. Period formula: $\frac{2\pi}{3}$ when $B=3$.

7. Identify the end behavior of $f(x)= -2\cdot 4^x+1$ as $x\to-\infty$.

   As $x\to-\infty$, $f(x)\to 1$. As $x$ decreases, $4^x$ approaches 0, so $f(x)$ approaches 1.

8. Identify the end behavior of $f(x)=\log_b(x)$ when $b>1$ as $x\to\infty$ and $x\to 0^+$.

   As $x\to\infty$, $f(x)\to\infty$; as $x\to 0^+$, $f(x)\to-\infty$. For $b>1$: grows without bound, decreases without bound near asymptote.

9. What transformation does $f(x)=\log_b(x-h)$ represent relative to $y=\log_b(x)$?

   Horizontal shift right by $h$ (left if $h<0$). Input transformation: subtracting shifts graph right.

10. State the general form for graphing a sinusoid with period, midline, and amplitude labeled.

    $y=A\sin(B(x-h))+k$ or $y=A\cos(B(x-h))+k$. Shows amplitude $A$, period factor $B$, shifts $h,k$.

11. What are the maximum and minimum values of $y=2\sin(x)-5$?

    Max $=-3$, Min $=-7$. Midline $-5$ plus/minus amplitude $2$.

12. Identify the period of $y=7\sin\left(\frac{\pi}{2}x\right)+3$.

    $4$. Period formula: $\frac{2\pi}{\pi/2}=4$ when $B=\frac{\pi}{2}$.

13. What is the general form of a logarithmic function used for graphing transformations?

    $f(x)=a\log_b(x-h)+k$ with $b>0$ and $b\neq 1$. Standard form showing all transformations with base restrictions.

14. What is the period of $y=A\sin(Bx)$ or $y=A\cos(Bx)$?

    Period: $\frac{2\pi}{|B|}$. Period formula: $\frac{2\pi}{|B|}$ for sine and cosine.

15. What is the inverse relationship between $y=b^x$ and $y=\log_b(x)$?

    They are inverses; their graphs reflect across $y=x$. Exponential and log functions undo each other.

16. State the five key points for one cycle of $y=\cos(x)$ on $[0,2\pi]$.

    $(0,1),(\frac{\pi}{2},0),(\pi,-1),(\frac{3\pi}{2},0),(2\pi,1)$. Cosine starts at maximum, crosses axis, reaches minimum, returns.

17. What is the domain of $f(x)=a\log_b(x-h)+k$?

    Domain: $(h,\infty)$. Argument must be positive, so $x>h$.

18. What is the range of $f(x)=a\log_b(x-h)+k$ if $a\neq 0$?

    Range: $(-\infty,\infty)$. Logarithmic functions have unrestricted output values.

19. Identify the end behavior of $f(x)=b^x$ when $b>1$ as $x\to\infty$ and $x\to-\infty$.

    As $x\to\infty$, $f(x)\to\infty$; as $x\to-\infty$, $f(x)\to 0$. For $b>1$: grows to infinity, approaches zero from left.

20. Identify the end behavior of $f(x)= -2\cdot 4^x+1$ as $x\to\infty$.

    As $x\to\infty$, $f(x)\to-\infty$. Negative coefficient with growing base: approaches negative infinity.

21. What is the midline of $y=2\sin(x)-5$?

    $y=-5$. Vertical shift parameter is $k=-5$.

22. What is the $x$-intercept of $f(x)=\log_b(x)$?

    $(1,0)$. When $x=1$, $\log_b(1)=0$, giving point $(1,0)$.

23. Find the $x$-intercept of $f(x)=\log_2(x)-3$.

    $(8,0)$. Set $f(x)=0$: $\log_2(x)-3=0$ gives $x=2^3=8$.

24. What is the amplitude of $y=A\sin(B(x-h))+k$ or $y=A\cos(B(x-h))+k$?

    Amplitude: $|A|$. Coefficient $A$ determines maximum distance from midline.

25. What is the midline of $y=A\sin(B(x-h))+k$ or $y=A\cos(B(x-h))+k$?

    Midline: $y=k$. Vertical shift $k$ determines center line of oscillation.

26. What is the domain of $f(x)=a\,b^{(x-h)}+k$?

    Domain: $(-\infty,\infty)$. Exponential functions are defined for all real numbers.

27. What is the period of $y=\sin(Bx)$ if $B$ is negative?

    Period: $\frac{2\pi}{|B|}$. Absolute value of $B$ gives same period regardless of sign.

28. What is the amplitude of $y=2\sin(x)-5$?

    $2$. Coefficient magnitude is $|2|=2$.

29. What is the period of $y=A\tan(Bx)$?

    Period: $\frac{\pi}{|B|}$. Tangent has period $\pi$, so formula is $\frac{\pi}{|B|}$.

30. What transformation does $f(x)=a\,b^x$ represent relative to $y=b^x$?

    Vertical stretch by $|a|$; reflection across $x$-axis if $a<0$. Coefficient affects vertical scaling and reflection.

31. What is the $y$-intercept of $f(x)=b^x$?

    $(0,1)$. When $x=0$, $b^0=1$, giving point $(0,1)$.

32. Find the horizontal asymptote of $f(x)=3\cdot 2^{x}-5$.

    $y=-5$. Vertical shift parameter $k=-5$ gives asymptote.

33. Identify whether $f(x)=5\cdot 1.2^x$ shows growth or decay.

    Growth. Base $1.2>1$ causes exponential growth.

34. Identify the end behavior of $f(x)=\log_b(x)$ when $0<b<1$ as $x\to\infty$ and $x\to 0^+$.

    As $x\to\infty$, $f(x)\to-\infty$; as $x\to 0^+$, $f(x)\to\infty$. For $0<b<1$: decreases without bound as $x$ increases.

35. What transformation does $f(x)=b^x+k$ represent relative to $y=b^x$?

    Vertical shift by $k$. Output transformation: adding shifts graph up.

36. Find the $y$-intercept of $f(x)=4\cdot 3^x-7$.

    $(0,-3)$. At $x=0$: $f(0)=4(1)-7=-3$.

37. What is the vertical asymptote of $f(x)=a\log_b(x-h)+k$?

    $x=h$. Horizontal shift parameter $h$ determines vertical asymptote.

38. Identify the midline of $y=7\sin\left(\frac{\pi}{2}x\right)+3$.

    $y=3$. Vertical shift parameter is $k=3$.

39. What is the horizontal asymptote of $f(x)=a\,b^{(x-h)}+k$?

    $y=k$. Vertical shift parameter $k$ determines horizontal asymptote.

40. What is the period of $y=\tan(5x)$?

    $\frac{\pi}{5}$. Tangent period formula: $\frac{\pi}{5}$ when $B=5$.

41. What is the midline of $y=\frac{1}{2}\cos(x)-1$?

    $y=-1$. Vertical shift parameter is $k=-1$.

42. What is the range of $y=A\sin(B(x-h))+k$?

    Range: $[k-|A|,\,k+|A|]$. Amplitude $|A|$ above and below midline $k$.

43. Find the $x$-intercept of $f(x)=2^x-8$.

    $(3,0)$. Set $f(x)=0$: $2^x=8=2^3$ gives $x=3$.

44. Identify the amplitude of $y=7\sin\left(\frac{\pi}{2}x\right)+3$.

    $7$. Coefficient magnitude is $|7|=7$.

45. What is the amplitude of $y=-3\sin(2x)+4$?

    $3$. Coefficient magnitude is $|-3|=3$.

46. What transformation does $f(x)=\log_b(x)+k$ represent relative to $y=\log_b(x)$?

    Vertical shift by $k$. Output transformation: adding shifts graph up.

47. Identify the end behavior of $f(x)=3\log_2(x-5)+1$ as $x\to\infty$.

    As $x\to\infty$, $f(x)\to\infty$. Positive coefficient with increasing argument: function grows without bound.

48. Identify the end behavior of $f(x)=b^x$ when $0<b<1$ as $x\to\infty$ and $x\to-\infty$.

    As $x\to\infty$, $f(x)\to 0$; as $x\to-\infty$, $f(x)\to\infty$. For $0<b<1$: decays to zero, grows from left.

49. What is the period of $y=2\cos\left(\frac{1}{4}x\right)$?

    $8\pi$. Period formula: $\frac{2\pi}{1/4}=8\pi$ when $B=\frac{1}{4}$.

50. Find the $y$-intercept of $f(x)=\log_5(x+1)$.

    $(0,0)$. At $x=0$: $f(0)=\log_5(1)=0$.

51. Find the vertical asymptote of $f(x)=2\log_3(x+4)-1$.

    $x=-4$. Horizontal shift parameter: $x+4=0$ gives $x=-4$.

52. What is the range of $y=A\cos(B(x-h))+k$?

    Range: $[k-|A|,\,k+|A|]$. Amplitude $|A|$ above and below midline $k$.

53. Identify whether $f(x)=\left(\frac{1}{3}\right)^x$ shows growth or decay.

    Decay. Base $\frac{1}{3}<1$ causes exponential decay.

54. State the five key points for one cycle of $y=\sin(x)$ on $[0,2\pi]$.

    $(0,0),(\frac{\pi}{2},1),(\pi,0),(\frac{3\pi}{2},-1),(2\pi,0)$. Sine starts at origin, peaks at $\frac{\pi}{2}$, completes cycle.

55. What is the range of $f(x)=a\,b^{(x-h)}+k$ if $a\neq 0$?

    Range: $(k,\infty)$ if $a>0$; $( -\infty,k)$ if $a<0$. Sign of $a$ determines if range extends above or below asymptote.

56. Identify the end behavior of $f(x)=3\log_2(x-5)+1$ as $x\to 5^+$.

    As $x\to 5^+$, $f(x)\to-\infty$. Positive coefficient: as $x$ approaches 5 from right, logarithm decreases.

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