Transformations of Functions - AP Precalculus

1. Identify the transformation: $f(x) = \frac{1}{x}$ to $f(x) = \frac{1}{-x}$

   Reflection across the y-axis. Negative input creates horizontal reflection.

2. What is the effect of $cf(x)$ where $0 < c < 1$?

   Vertical compression by a factor of $c$. Multiplying by factor less than 1 compresses vertically.

3. What is the effect of $f(x)$ to $f(x + c)$ on its domain?

   Subtract $c$ from each element of the domain. Horizontal shift adjusts input values by constant.

4. Identify the transformation: $f(x) = e^x$ to $f(x) = e^{-x}$

   Reflection across the y-axis. Negative exponent reflects exponential across $y$-axis.

5. What transformation does $-f(x)$ represent?

   Reflection across the x-axis. Negative sign flips graph over the horizontal axis.

6. What is the effect of $cf(x)$ where $c > 1$?

   Vertical stretch by a factor of $c$. Multiplying by factor greater than 1 stretches vertically.

7. Identify the transformation: $f(x) = x^3$ to $f(x) = x^3 - 2$

   Vertical shift downward by 2 units. Subtracting 2 from output moves graph down.

8. Identify the transformation: $f(x) = \frac{1}{x}$ to $f(x) = \frac{1}{x} + 1$

   Vertical shift upward by 1 unit. Adding 1 to function output shifts graph upward.

9. What transformation is applied: $f(x) = \frac{1}{x}$ to $f(x) = \frac{1}{(x-4)}$?

   Horizontal shift right by 4 units. Subtracting 4 from input shifts right by 4.

10. What is the effect of $f(x - c)$ on the graph of $f(x)$?

    Horizontal shift right by $c$ units. Subtracting from input moves graph in same direction horizontally.

11. What is the effect of $f(x)$ to $f(-x)$ on its domain?

    Negate each element of the domain. Reflection changes signs of all input values.

12. Identify the transformation: $f(x) = \frac{1}{x}$ to $f(x) = \frac{1}{x} + 7$

    Vertical shift upward by 7 units. Adding 7 to function output shifts upward.

13. What is the effect of $f(x) - c$ on the graph of $f(x)$?

    Vertical shift downward by $c$ units. Subtracting constant from function output moves graph down.

14. What transformation does $f(-x)$ represent?

    Reflection across the y-axis. Negative input creates mirror image across vertical axis.

15. Identify the transformation: $f(x) = x^2$ to $f(x) = x^2 + 5$

    Vertical shift upward by 5 units. Adding 5 to function output shifts graph upward.

16. What transformation does $f(x) = x^2$ to $f(x) = -x^2$ represent?

    Reflection across the x-axis. Negative sign flips parabola across $x$-axis.

17. Identify the transformation: $f(x) = |x|$ to $f(x) = 3|x|$

    Vertical stretch by a factor of 3. Factor 3 multiplies all $y$-values by 3.

18. What is the effect of $f(x)$ to $-f(x)$ on its range?

    Negate each element of the range. Reflection flips signs of all output values.

19. What transformation does $f(x) = x^2$ to $f(x) = (x + 5)^2$ represent?

    Horizontal shift left by 5 units. Adding 5 to input shifts left by 5 units.

20. Identify the transformation: $f(x) = \frac{1}{x}$ to $f(x) = \frac{1}{x} - 5$

    Vertical shift downward by 5 units. Subtracting 5 from output moves graph downward.

21. Identify the transformation: $f(x) = x^2$ to $f(x) = (x - 3)^2$

    Horizontal shift right by 3 units. Inside parentheses: subtract 3 from $x$ shifts right.

22. What is the effect of $f(x)$ to $cf(x)$ where $c < 0$?

    Vertical stretch/compression and reflection across x-axis. Negative coefficient reflects and scales vertically.

23. Identify the transformation: $f(x) = \frac{1}{x}$ to $f(x) = \frac{1}{x} - 3$

    Vertical shift downward by 3 units. Subtracting 3 from function output shifts downward.

24. Identify the transformation: $f(x) = x^3$ to $f(x) = (x + 4)^3$

    Horizontal shift left by 4 units. Adding 4 to input moves graph left by 4.

25. Identify the transformation: $f(x) = \frac{1}{x}$ to $f(x) = \frac{1}{x+2}$

    Horizontal shift left by 2 units. Adding 2 to input shifts graph left by 2.

26. What is the effect of $f(x + c)$ on the graph of $f(x)$?

    Horizontal shift left by $c$ units. Adding to input moves graph opposite direction horizontally.

27. What transformation does $f(x) = x^2$ to $f(x) = (x - 6)^2$ represent?

    Horizontal shift right by 6 units. Subtracting 6 from input shifts right by 6.

28. Identify the transformation: $f(x) = |x|$ to $f(x) = |x - 1|$

    Horizontal shift right by 1 unit. Subtracting 1 from input shifts right by 1.

29. Identify the transformation: $f(x) = |x|$ to $f(x) = |x + 2|$

    Horizontal shift left by 2 units. Adding 2 to input shifts left by 2 units.

30. What is the effect of $f(cx)$ where $0 < c < 1$?

    Horizontal stretch by a factor of $\frac{1}{c}$. Input coefficient less than 1 stretches horizontally.

31. What is the effect of $f(x)$ to $f(x) + c$ on its range?

    Add $c$ to each element of the range. Vertical shift adds constant to all output values.

32. What is the effect of $f(x) + c$ on the graph of $f(x)$?

    Vertical shift upward by $c$ units. Adding constant to function output moves graph up.

33. Identify the transformation: $f(x) = |x|$ to $f(x) = 2|x|$

    Vertical stretch by a factor of 2. Coefficient 2 multiplies all $y$-values by 2.

34. What is the combined effect of $-f(x + 2)$?

    Reflection across x-axis and left shift by 2. Combines vertical reflection and horizontal translation.

35. Identify the transformation of $f(x) = x^2$ to $f(x) = 2x^2 + 3$.

    Vertical stretch by 2 and shift up by 3. Multiplies output by 2, then adds 3 to result.

36. What is the effect of $f(cx)$ where $c > 1$?

    Horizontal compression by a factor of $\frac{1}{c}$. Input coefficient greater than 1 compresses horizontally.

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