The Big Picture
A transformation changes the position, size, direction, or steepness of a graph. For linear functions, these changes are especially easy to see because the graph remains a straight line.
General Transformation Form
Each letter controls a different graph transformation.
Start with the Parent Linear Function
The simplest linear parent function is f(x) = x. Its graph is a straight line passing through the origin with slope 1. Every transformation below begins with this basic graph.
The line rises 1 unit for every 1 unit moved to the right.
Vertical Translations: Moving Up and Down
1. Shift Up by k Units
Adding a positive number outside the function moves every point upward.
y = f(x) + 3 = x + 3
The graph moves up 3 units. Its slope does not change.
2. Shift Down by k Units
Subtracting a positive number outside the function moves every point downward.
y = f(x) − 4 = x − 4
The graph moves down 4 units while keeping the same slope.
Horizontal Translations: Moving Left and Right
3. Shift Right by h Units
A number inside the function has the opposite visual direction. Replacing x with x − 2 shifts the graph right 2 units.
y = f(x − 2) = x − 2
4. Shift Left by h Units
Replacing x with x + 5 shifts the graph left 5 units.
y = f(x + 5) = x + 5
Memory tip: Outside the function, the sign behaves normally. Inside the function, the direction appears reversed: x − h moves right, while x + h moves left.
Vertical Scaling and Reflection
| Transformation | Equation | What happens? |
|---|---|---|
| Vertical stretch | y = 3f(x) | Every y-value is tripled. |
| Vertical compression | y = 0.5f(x) | Every y-value is halved. |
| Reflection across x-axis | y = −f(x) | All y-values change sign. |
On a phone, swipe left or right inside the table if needed.
Horizontal Scaling and Reflection
Changes inside the function affect the x-values. The visual effect is often the reciprocal of the number you see.
5. Horizontal Compression
For y = f(2x), the graph is compressed horizontally by a factor of 1/2. Points reach the same y-values in half the horizontal distance.
6. Horizontal Stretch
For y = f(0.5x), the graph is stretched horizontally by a factor of 2.
7. Reflection across the y-axis
Replacing x with −x reflects the graph across the y-axis:
y = f(−x)
Combined Transformations
Several transformations can appear in one equation. Consider:
Read this equation from the inside outward. The graph shifts right 3 units, then every y-value is doubled, and finally the graph shifts up 4 units.
Step 1: x − 3 means shift right 3 units.
Step 2: The 2 outside the function creates a vertical stretch by factor 2.
Step 3: +4 shifts the graph up 4 units.
Transforming a Linear Equation Directly
Suppose the original function is f(x) = 2x + 1. A transformed version might be:
The new line is shifted right 2 units and up 5 units. Its slope remains 2, so the lines are parallel.
Common Mistakes to Avoid
Inside versus outside: Remember that x − h shifts right, not left.
Horizontal factors: The number inside the function has a reciprocal effect on horizontal size.
Order of operations: Handle the inside shift first, then the vertical scale, and then the outside shift.
Slope confusion: A vertical shift changes the intercept but does not change the slope.
Practice Problems
Identify the transformation in each equation:
1. y = f(x) + 6
2. y = f(x − 4)
3. y = −2f(x)
4. y = f(3x)
5. y = 0.5f(x + 2) − 1
Tap to reveal the answers
Final Takeaway
For linear-function transformations, remember the controls: h moves the graph left or right, k moves it up or down, a changes vertical size and reflection, and b changes horizontal size and reflection. Read the equation from the inside outward to avoid sign mistakes.
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