Function Transformations
By the end of this lesson, you’ll be able to:
- Describe how graphs shift, reflect, stretch, and compress.
- Apply transformation rules directly to function formulas.
- Identify multiple transformations within a single function.
- Match transformed graphs to their parent (base) functions.
Key Ideas
Transformations tell us how a graph moves or changes shape relative to its parent function \(f(x)\).
A parent function is the simplest version of a function family.
Common parent functions include:
| Parent Function | Graph Shape |
|---|---|
| \(y=x\) | Line |
| \(y=x^2\) | Parabola |
| \(y=|x|\) | V-shape |
| \(y=\sqrt{x}\) | Square root curve |
| \(y=x^3\) | Cubic curve |
| \(y=b^x\) | Exponential curve |
Transformations change the graph without changing the basic family.
| Transformation | Rule | Effect |
|---|---|---|
| Vertical shift up | \(f(x)+k\) | Up \(k\) units |
| Vertical shift down | \(f(x)-k\) | Down \(k\) units |
| Horizontal shift right | \(f(x-h)\) | Right \(h\) units |
| Horizontal shift left | \(f(x+h)\) | Left \(h\) units |
| Reflect over \(x\)-axis | \(-f(x)\) | Flip vertically |
| Reflect over \(y\)-axis | \(f(-x)\) | Flip horizontally |
| Vertical stretch | \(af(x),\ |a|>1\) | Narrower / steeper |
| Vertical compression | \(af(x),\ 0<|a|<1\) | Wider / flatter |
| Horizontal compression | \(f(bx),\ |b|>1\) | Narrower |
| Horizontal stretch | \(f(bx),\ 0<|b|<1\) | Wider |
The most important idea:
- Changes outside the function affect the graph vertically.
- Changes inside the function affect the graph horizontally.
For example:
\[ y=f(x)+4 \]
moves the graph up 4 units.
But
\[ y=f(x+4) \]
moves the graph left 4 units.
Common Problem Types
1. Vertical Shifts
A number added or subtracted outside the function shifts the graph up or down.
Example:
\[ y=x^2+5 \]
This shifts the graph of \(y=x^2\) up 5 units.
Example:
\[ y=|x|-3 \]
This shifts the graph of \(y=|x|\) down 3 units.
2. Horizontal Shifts
A number added or subtracted inside the function shifts the graph left or right.
Example:
\[ y=(x-4)^2 \]
This shifts the graph of \(y=x^2\) right 4 units.
Example:
\[ y=\sqrt{x+2} \]
This shifts the graph of \(y=\sqrt{x}\) left 2 units.
Horizontal shifts feel backward:
- \(f(x-h)\) moves right \(h\) units.
- \(f(x+h)\) moves left \(h\) units.
3. Reflections
A negative sign reflects a graph.
If the negative is outside the function, the graph reflects across the \(x\)-axis.
Example:
\[ y=-x^2 \]
This reflects the graph of \(y=x^2\) across the \(x\)-axis.
If the negative is inside the function, the graph reflects across the \(y\)-axis.
Example:
\[ y=f(-x) \]
This reflects the graph of \(y=f(x)\) across the \(y\)-axis.
4. Stretches and Compressions
A coefficient changes the steepness or width of a graph.
Example:
\[ y=3x^2 \]
This is a vertical stretch by a factor of 3.
Example:
\[ y=\frac12x^2 \]
This is a vertical compression by a factor of \(\frac12\).
For many SAT questions, vertical stretches and compressions are more common than horizontal stretches and compressions.
5. Combined Transformations
Many problems combine several transformations in one equation.
Example:
\[ y=-2(x-3)^2+5 \]
Start with the parent function:
\[ y=x^2 \]
Then identify each transformation:
- \((x-3)\) shifts the graph right 3 units.
- The coefficient \(-2\) reflects the graph across the \(x\)-axis.
- The coefficient \(2\) vertically stretches the graph by a factor of 2.
- The \(+5\) shifts the graph up 5 units.
So the graph is:
- shifted right 3 units
- shifted up 5 units
- reflected across the \(x\)-axis
- vertically stretched by a factor of 2
Strategies
- First identify the parent function.
- Look inside parentheses for horizontal changes.
- Look outside the function for vertical changes.
- A negative outside the function reflects across the \(x\)-axis.
- A negative inside the function reflects across the \(y\)-axis.
- Coefficients greater than 1 create stretches.
- Coefficients between 0 and 1 create compressions.
- For combined transformations, list each change separately.
A useful checklist:
- What is the parent function?
- Is there a horizontal shift?
- Is there a vertical shift?
- Is there a reflection?
- Is there a stretch or compression?
Worked Examples
Example 1 — Vertical Shift
Describe the transformation:
\[ g(x)=x^2+3 \]
The parent function is:
\[ y=x^2 \]
The \(+3\) is outside the function, so the graph shifts up 3 units.
Answer: Up 3 units.
Example 2 — Horizontal Shift
Describe the transformation:
\[ y=(x-4)^2 \]
The parent function is:
\[ y=x^2 \]
The expression \((x-4)\) shifts the graph right 4 units.
Answer: Right 4 units.
Example 3 — Reflection Across the \(x\)-Axis
Describe the transformation:
\[ h(x)=-\sqrt{x} \]
The parent function is:
\[ y=\sqrt{x} \]
The negative sign outside the function reflects the graph across the \(x\)-axis.
Answer: Reflection across the \(x\)-axis.
Example 4 — Combined Transformations
Describe the transformations of:
\[ y=-3(x+2)^2-4 \]
The parent function is:
\[ y=x^2 \]
Now identify each transformation:
- \((x+2)\) shifts the graph left 2 units.
- The negative sign outside reflects the graph across the \(x\)-axis.
- The coefficient \(3\) vertically stretches the graph by a factor of 3.
- The \(-4\) shifts the graph down 4 units.
Answer:
- left 2 units
- down 4 units
- reflected across the \(x\)-axis
- vertically stretched by a factor of 3
Example 5 — Absolute Value Combined Transformations
Describe the transformations of:
\[ y=-|x-1|+2 \]
The parent function is:
\[ y=|x| \]
Now identify each transformation:
- \((x-1)\) shifts the graph right 1 unit.
- The negative sign outside reflects the graph across the \(x\)-axis.
- The \(+2\) shifts the graph up 2 units.
Answer:
- right 1 unit
- up 2 units
- reflected across the \(x\)-axis
Example 6 — Square Root Combined Transformations
Describe the transformations of:
\[ y=4\sqrt{x+1} \]
The parent function is:
\[ y=\sqrt{x} \]
Now identify each transformation:
- \((x+1)\) shifts the graph left 1 unit.
- The coefficient \(4\) vertically stretches the graph by a factor of 4.
Answer:
- left 1 unit
- vertically stretched by a factor of 4
Common Mistakes
- Confusing horizontal and vertical shifts.
- Forgetting that \(f(x-h)\) moves right, not left.
- Mixing up reflections over the \(x\)-axis and \(y\)-axis.
- Missing a stretch or compression when multiple transformations are present.
- Identifying only one transformation when the equation contains several.
- Forgetting to compare the transformed graph to the parent function.
Practice Problems
- Describe the shift: \(y=x^2-5\).
- Describe the transformation: \(y=-|x|\).
- Describe the transformation: \(y=\sqrt{x+2}\).
- Does \(y=3f(x)\) stretch, compress, or shift?
- What does \(y=f(-x)\) do to the graph?
- Describe all transformations of:
\[ y=-2(x-4)^2+1 \]
- Describe all transformations of:
\[ y=\frac12(x+3)^2-6 \]
- Describe all transformations of:
\[ y=-|x-5|-2 \]
- Describe all transformations of:
\[ y=4\sqrt{x+1} \]
- Describe all transformations of:
\[ y=-(x+6)^2+8 \]
1.
\[ y=x^2-5 \]
The \(-5\) shifts the graph down 5 units.
Answer: Down 5 units.
2.
\[ y=-|x| \]
The negative sign outside reflects the graph across the \(x\)-axis.
Answer: Reflection across the \(x\)-axis.
3.
\[ y=\sqrt{x+2} \]
The \(+2\) inside shifts the graph left 2 units.
Answer: Left 2 units.
4.
\[ y=3f(x) \]
The coefficient 3 outside the function creates a vertical stretch.
Answer: Vertical stretch by factor 3.
5.
\[ y=f(-x) \]
The negative sign inside reflects the graph across the \(y\)-axis.
Answer: Reflection across the \(y\)-axis.
6.
\[ y=-2(x-4)^2+1 \]
Transformations:
- right 4 units
- up 1 unit
- reflection across the \(x\)-axis
- vertical stretch by factor 2
7.
\[ y=\frac12(x+3)^2-6 \]
Transformations:
- left 3 units
- down 6 units
- vertical compression by factor \(\frac12\)
8.
\[ y=-|x-5|-2 \]
Transformations:
- right 5 units
- down 2 units
- reflection across the \(x\)-axis
9.
\[ y=4\sqrt{x+1} \]
Transformations:
- left 1 unit
- vertical stretch by factor 4
10.
\[ y=-(x+6)^2+8 \]
Transformations:
- left 6 units
- up 8 units
- reflection across the \(x\)-axis
Summary
- Transformations describe how a graph changes from its parent function.
- Vertical changes occur outside the function.
- Horizontal changes occur inside the function.
- Shifts move graphs without changing their shape.
- Reflections flip graphs across an axis.
- Stretches and compressions change steepness or width.
- Combined transformations include multiple changes in the same equation.
- SAT questions often ask you to identify several transformations at once.
- Inside → horizontal.
- Outside → vertical.
- \(f(x-h)\) moves right.
- \(f(x+h)\) moves left.
- Negative outside → reflect over the \(x\)-axis.
- Negative inside → reflect over the \(y\)-axis.
- Larger outside coefficient → vertical stretch.
- Fraction outside coefficient → vertical compression.
- Identify transformations one at a time.