The SAT tests whether you can look at and predict what happens to the graph when it changes to or . Here's your complete guide.
The Transformation Rules
Starting with :
| Transformation | Effect on Graph |
|---|---|
| Shift up units | |
| Shift down units | |
| Shift left units | |
| Shift right units | |
| Reflect over the x-axis | |
| Reflect over the y-axis | |
| where | Vertical stretch |
| where | Vertical compression |
The counterintuitive part: horizontal shifts go in the opposite direction of the sign.
Vertical Shifts
Example 1: If , then shifts the parabola up 4 units. Every point moves from to .
Example 2: shifts the parabola down 2 units.
Horizontal Shifts
Example 3: shifts the parabola right 3 units.
Why right? Because — you need a bigger to get the same output.
Example 4: shifts left 5 units (since ).
Reflections
Example 5: If , then flips the graph over the x-axis (all y-values become negative).
Example 6: flips the graph over the y-axis (the graph now extends to the left).
Vertical Stretches and Compressions
Example 7: stretches vertically by a factor of 3 (every y-value triples).
Example 8: compresses vertically by a factor of (every y-value halves).
Combining Transformations
Example 9: Describe the transformation from to .
- Shift right 1 unit (the )
- Vertical stretch by factor 2 (the coefficient)
- Shift up 3 units (the )
The vertex moves from to .
SAT-Style Question
Example 10: The graph of passes through the point . Which point must be on the graph of ?
Shift right 4: -coordinate becomes
Shift up 1: -coordinate becomes
Answer:
Practice Problems
Problem 1: The graph of contains the point . What point is on ?
Solution
Shift left 2:
Problem 2: How does the graph of compare to ?
Solution
Reflected over the x-axis, shifted right 4, shifted up 7. Vertex at opening downward.
Problem 3: If , what is the value of ?
Solution
Key Takeaways
- Inside the function (with ): horizontal changes go in the opposite direction
- Outside the function: vertical changes go in the expected direction
- Reflections: flips over x-axis; flips over y-axis
- Read transformations from the inside out when they're combined
- On the SAT, apply transformations to specific points — it's faster than analyzing the whole graph
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