Transformations of functions on Khan Academy describe how changing an equation reshapes its graph. Learners explore shifts, stretches, and flips that connect algebraic changes to visual patterns.
This structured approach helps students predict graph behavior by analyzing equation modifications step by step. The interactive exercises provide immediate feedback, reinforcing each transformation concept through practice.
| Transformation Type | Equation Change | Graph Effect | Example |
|---|---|---|---|
| Vertical Shift | f(x) + k | Moves up or down | f(x) + 3 shifts up 3 units |
| Horizontal Shift | f(x − h) | Moves left or right | f(x − 2) shifts right 2 units |
| Vertical Stretch/Compression | a·f(x) | Stretches or compresses vertically | 2f(x) stretches vertically by factor 2 |
| Reflection | −f(x) or f(−x) | Flips over axis | −f(x) reflects over x-axis |
Understanding Vertical and Horizontal Shifts
Vertical shifts move a graph up or down without altering its shape. Adding a positive constant shifts the graph upward, while subtracting shifts it downward.
Horizontal shifts translate the graph left or right along the x-axis. These shifts inside the function argument work in the opposite direction of the sign, which often surprises learners at first.
Exploring Stretches and Compressions
Vertical stretches and compressions change the steepness of the graph. Multiplying the function by a number greater than 1 makes it taller, while a factor between 0 and 1 makes it shorter.
When the multiplier is negative, the graph also reflects over the x-axis. Horizontal stretches and compressions apply to the input variable and require careful attention to the coefficient inside the function.
Mastering Reflections Across Axes
Multiplying the entire function by −1 reflects the graph over the x-axis, flipping it upside down. This transformation changes the sign of all y-values while keeping x-values the same.
Replacing x with −x reflects the graph over the y-axis, creating a mirror image left to right. Learners practice identifying both reflections across axes in coordinate plane exercises.
Combining Multiple Transformations
Real functions often involve several transformations at once. The order of operations matters, especially when stretches, compressions, and shifts are combined in a single equation.
Khan Academy provides step-by-step guided problems that break down complex transformations into manageable stages. Students learn to identify key points on graphs after each modification to track the overall change.
Applying Transformations to Real Graphs
Practice builds intuition for how each algebraic change appears visually on coordinate axes. Consistent exercises help students recognize transformation patterns quickly and accurately.
- Identify the base function before any transformations are applied.
- Track vertical shifts by observing constant terms added outside the function.
- Note horizontal shifts by examining changes inside the function argument.
- Detect stretches, compressions, and reflections through coefficients multiplying the function or its input.
- Combine transformations step by step, respecting the correct order of operations.
FAQ
Reader questions
How do I know whether to shift left or right when the sign inside the function changes?
Subtracting h from x shifts the graph right by h units, while adding h shifts it left by h units, because the transformation acts in the opposite direction of the sign.
Can a vertical stretch also cause a reflection in the same step?
Yes, if you multiply the function by a negative number, the vertical stretch or compression is accompanied by a reflection over the x-axis.
What happens to the point (0, 0) after a horizontal shift and vertical shift?
The point (0, 0) moves to (h, k) when the function transforms to f(x − h) + k, following the direction and magnitude of each shift.
Why does the coefficient inside the function affect the graph horizontally instead of vertically?
Changes to the input variable x affect the horizontal position because they determine when the function reaches specific output values, reversing the intuitive left-right direction.