Transformations of Functions and Graphs - Algebra 2

1. What is the graph effect of replacing $f(x)$ with $f(x)+k$ for $k>0$?

   Shift the graph up $k$ units. Adding $k$ to the function output translates all points vertically upward.

2. What is the graph effect of replacing $f(x)$ with $k f(x)$ for $k>1$?

   Vertical stretch by factor $k$. Multiplying by $k>1$ stretches all $y$-values away from the $x$-axis.

3. What is the effect of $g(x)=f(1\cdot x)$ on the graph of $f$?

   No change. Multiplying input by one produces an identical transformation.

4. What is the point-mapping rule for $g(x)=f(x+k)$ from a point $(x,y)$ on $f$?

   $(x,y)\to(x-k,y)$. Horizontal shift moves each point leftward by $k$ units.

5. What is the graph effect of replacing $f(x)$ with $f(x-k)$ for $k<0$?

   Shift the graph left $|k|$ units. Subtracting negative $k$ from input shifts the graph leftward.

6. What graph symmetry identifies an even function?

   Symmetry about the $y$-axis. Even functions have mirror symmetry across the $y$-axis.

7. What is the effect of replacing $f(x)$ with $f(-x)$?

   Reflect the graph across the $y$-axis. Negative input creates a mirror image across the $y$-axis.

8. Identify the $y$-intercept of $g(x)=f(x+k)$ in terms of $f$.

   $g(0)=f(k)$. The $y$-intercept becomes $f(k)$.

9. What is the graph effect of replacing $f(x)$ with $k f(x)$ for $0<k<1$?

   Vertical compression by factor $k$. Multiplying by $0<k<1$ compresses all $y$-values toward the $x$-axis.

10. Find $k$ if $g(x)=f(x+k)$ and a point $(5,7)$ on $f$ becomes $(2,7)$ on $g$.

    $k=3$. Point $(5,7)$ moves left to $(2,7)$, so $k=5-2=3$.

11. What is the graph effect of replacing $f(x)$ with $f(x)+k$ for $k<0$?

    Shift the graph down $|k|$ units. Adding negative $k$ translates all points downward by the magnitude.

12. What is the graph effect of replacing $f(x)$ with $f(kx)$ for $k<0$?

    Reflect across $y$-axis and scale horizontally by $|k|$. Negative input coefficient reflects across $y$-axis and scales horizontally.

13. Find $k$ if $g(x)=k f(x)$ and a point $(4,3)$ on $f$ becomes $(4,-12)$ on $g$.

    $k=-4$. The scaling factor is $\frac{-12}{3}=-4$.

14. Identify whether $f(x)=\frac{1}{x}$ is even, odd, or neither.

    Odd. Reciprocal function has origin symmetry: $\frac{1}{-x}=-\frac{1}{x}$.

15. What is the point-mapping rule for $g(x)=f(kx)$ from a point $(x,y)$ on $f$ with $k\neq 0$?

    $(x,y)\to\left(\frac{x}{k},y\right)$. Horizontal scaling divides each $x$-coordinate by $k$.

16. What is the defining equation for an odd function?

    $f(-x)=-f(x)$ for all $x$ in the domain. This equation defines symmetry about the origin.

17. Identify whether $f(x)=x^3+x^2$ is even, odd, or neither.

    Neither. Mix of even and odd powers breaks symmetry properties.

18. What is the defining equation for an even function?

    $f(-x)=f(x)$ for all $x$ in the domain. This equation defines symmetry about the $y$-axis.

19. What is the horizontal scale factor for $g(x)=f(4x)$ relative to $f(x)$?

    Horizontal compression by factor $4$. Input coefficient $4$ creates horizontal compression by factor $4$.

20. What is the graph effect of replacing $f(x)$ with $f(x+k)$ for $k<0$?

    Shift the graph right $|k|$ units. Adding negative $k$ to input shifts the graph rightward.

21. Identify the $y$-intercept of $g(x)=f(x)+k$ in terms of $f(0)$.

    $g(0)=f(0)+k$. The $y$-intercept is the sum of original intercept and $k$.

22. What is the point-mapping rule for $g(x)=k f(x)$ from a point $(x,y)$ on $f$?

    $(x,y)\to(x,ky)$. Vertical scaling multiplies each $y$-coordinate by $k$.

23. Identify whether $f(x)=x^4-3x^2+7$ is even, odd, or neither.

    Even. All terms have even powers, so $f(-x)=f(x)$.

24. What is the effect on even/odd status when forming $g(x)=f(x)+k$ with $k\neq 0$?

    It generally becomes neither even nor odd. Vertical shifts break the symmetry through origin or $y$-axis.

25. Identify the effect of $g(x)=f(-x)+5$ on the graph relative to $f(x)$.

    Reflect across $y$-axis, then shift up $5$. First reflect horizontally, then translate vertically.

26. What is the graph effect of replacing $f(x)$ with $f(x-k)$ for $k>0$?

    Shift the graph right $k$ units. Subtracting $k$ from input shifts the graph rightward by $k$ units.

27. What is the point-mapping rule for $g(x)=f(x-k)$ from a point $(x,y)$ on $f$?

    $(x,y)\to(x+k,y)$. Horizontal shift moves each point rightward by $k$ units.

28. What is the graph effect of replacing $f(x)$ with $k f(x)$ for $k<0$?

    Reflect across $x$-axis and scale by $|k|$. Negative coefficient flips across $x$-axis and scales by absolute value.

29. Identify whether $f(x)=x^3-5x$ is even, odd, or neither.

    Odd. All terms have odd powers, so $f(-x)=-f(x)$.

30. Identify the $y$-intercept of $g(x)=k f(x)$ in terms of $f(0)$.

    $g(0)=k f(0)$. The $y$-intercept is $k$ times the original intercept.

31. Find $k$ if $g(x)=f(kx)$ and a point $(6,2)$ on $f$ becomes $(3,2)$ on $g$.

    $k=2$. If $(6,2)$ becomes $(3,2)$, then $\frac{6}{k}=3$, so $k=2$.

32. What is the graph effect of replacing $f(x)$ with $f(kx)$ for $k>1$?

    Horizontal compression by factor $k$. Multiplying input by $k>1$ compresses the graph horizontally.

33. What is the graph effect of replacing $f(x)$ with $f(x+k)$ for $k>0$?

    Shift the graph left $k$ units. Adding $k$ to input shifts the graph leftward by $k$ units.

34. What is the effect of replacing $f(x)$ with $-f(x)$?

    Reflect the graph across the $x$-axis. Negative coefficient flips all $y$-values to opposite signs.

35. What must be true about the domain to classify a function as even or odd?

    If $x$ is in the domain, then $-x$ must also be in the domain. The domain must be symmetric about zero for classification.

36. Find $k$ if $g(x)=f(x)+k$ and the $y$-intercept changes from $2$ to $-1$.

    $k=-3$. Change in $y$-intercept equals the vertical shift amount.

37. What is the effect of replacing $f(x)$ with $-f(-x)$?

    Reflect across both axes (equivalently rotate $180^{\circ}$). Combines both reflections, equivalent to point reflection through origin.

38. What is the horizontal scale factor for $g(x)=f(\frac{1}{3}x)$ relative to $f(x)$?

    Horizontal stretch by factor $3$. Input coefficient $\frac{1}{3}$ creates horizontal stretch by factor $3$.

39. What is the effect on even/odd status when forming $g(x)=k f(x)$ with $k\neq 0$?

    Even/odd status stays the same as $f$. Vertical scaling preserves the original symmetry pattern.

40. Identify the effect of $g(x)=f(x-4)+1$ on the graph relative to $f(x)$.

    Shift right $4$, then shift up $1$. First shift horizontally, then translate vertically.

41. What is the graph effect of replacing $f(x)$ with $f(kx)$ for $0<k<1$?

    Horizontal stretch by factor $\frac{1}{k}$. Multiplying input by $0<k<1$ stretches the graph horizontally.

42. Identify the effect of $g(x)=-3f(\frac{1}{2}x)$ on the graph relative to $f(x)$.

    Horizontal stretch by $2$; reflect $x$-axis; vertical stretch by $3$. Combines horizontal stretch, $x$-axis reflection, and vertical stretch.

43. Identify whether $f(x)=|x|$ is even, odd, or neither.

    Even. Absolute value creates $y$-axis symmetry: $|-x|=|x|$.

44. What is the effect of $g(x)=1\cdot f(x)$ on the graph of $f$?

    No change. Multiplying by one produces an identical transformation.

45. What is the effect of $g(x)=f(x)+0$ on the graph of $f$?

    No change. Adding zero produces an identical transformation.

46. Find $k$ if $g(x)=f(x)+k$ and a point $(1,-2)$ on $f$ becomes $(1,5)$ on $g$.

    $k=7$. Vertical shift from $-2$ to $5$ requires adding $7$.

47. Identify the $y$-intercept of $g(x)=f(kx)$ in terms of $f(0)$.

    $g(0)=f(0)$. Horizontal scaling doesn't affect the $y$-intercept at $x=0$.

48. Identify whether $f(x)=\frac{1}{x^2}$ is even, odd, or neither.

    Even. Even power in denominator creates $y$-axis symmetry.

49. Find $k$ if $g(x)=k f(x)$ and a point $(-3,8)$ on $f$ becomes $(-3,2)$ on $g$.

    $k=\frac{1}{4}$. Scaling factor is $\frac{2}{8}=\frac{1}{4}$.

50. What graph symmetry identifies an odd function?

    Symmetry about the origin (a $180^{\circ}$ rotation). Odd functions have point symmetry through the origin.

51. Identify the effect of $g(x)=2f(x)-3$ on the graph relative to $f(x)$.

    Vertical stretch by $2$, then shift down $3$. First stretch vertically, then translate downward.

52. What is the point-mapping rule for $g(x)=f(x)+k$ from a point $(x,y)$ on $f$?

    $(x,y)\to(x,y+k)$. Vertical shift adds $k$ to each $y$-coordinate.

53. Find $k$ if $g(x)=f(kx)$ and a point $(-4,9)$ on $f$ becomes $(2,9)$ on $g$.

    $k=-2$. If $(-4,9)$ becomes $(2,9)$, then $\frac{-4}{k}=2$, so $k=-2$.

Study these flashcards