Unit 1 Study Guide: Function Transformations

SOL Standard: AFDA.AF.1

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Date: _______________

Key Concepts Summary (The Rules)

Parent Functions: Linear ($f(x)=x$), Quadratic ($f(x)=x^2$), Exponential ($f(x)=2^x$).

The "Master Equation" for transformations is $f(x) = a \cdot \text{parent}(x-h) + k$.

The value of $a$ (in front) tells you two things:
- If it's negative, the graph reflects (flips) over the x-axis.
- If its absolute value is $> 1$, it's a vertical stretch (skinnier).
- If its absolute value is between 0 and 1, it's a vertical compression (wider).
The value of $h$ (inside, with $x$) tells you the horizontal shift. Remember it's the opposite of the sign! $(x-h)$ moves right, $(x+h)$ moves left.
The value of $k$ (at the end) tells you the vertical shift. It moves up for '+' and down for '-'.

WORKED EXAMPLE:

$f(x) = -3(x+1)^2 - 4$

Answer: Reflects over x-axis, Vertical Stretch by 3, Left 1, Down 4.

1. $f(x) = (x+2)^2 - 3$

2. $f(x) = -x^2 + 5$

3. $f(x) = 3(x-1)^2$

4. $f(x) = \frac{1}{2}(x)^2 + 8$

5. $f(x) = -2(x-4)^2 + 1$

6. $f(x) = 4(x+5)^2$

7. $f(x) = -(x)-3$

8. $f(x) = 2^x - 7$

WORKED EXAMPLE:

A quadratic function is reflected over the x-axis and moved left 2.

Answer: $f(x) = -(x+2)^2$

9. A quadratic function is moved right 5 units and down 2 units.

10. A quadratic function is reflected over the x-axis and moved up 9 units.

11. A quadratic function is vertically stretched by a factor of 4.

12. A quadratic function is vertically compressed by $\frac{1}{3}$, moved left 1, and up 6.

13. A linear function is stretched by 2 and moved down 5.

14. An exponential function is reflected over the x-axis and moved right 3.

WORKED EXAMPLE:

A quadratic graph's vertex is at $(2, 1)$. It opens upwards normally. The vertex of the parent is at $(0,0)$. So it moved right 2 and up 1.

Answer: $f(x) = (x-2)^2 + 1$

15. Equation for Graph A (Quadratic):

16. Equation for Graph B (Linear):

17. Equation for Graph C (Quadratic):

18. Equation for Graph D (Exponential):

19. Equation for Graph E (Quadratic):

20. Equation for Graph F (Linear):