To make a graph move horizontally, you apply transformations to its equation. Specifically, you modify the x variable. The type of modification determines the direction and extent of the horizontal shift.
Horizontal Shifts Explained
According to the provided reference, horizontal shifts are achieved by adding or subtracting a constant value from the x variable within the function's equation.
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Shifting to the Left: To shift a graph to the left, you add a positive constant, a, to the x variable. The equation becomes:
y = f(x + a) -
Shifting to the Right: To shift a graph to the right, you subtract a positive constant, a, from the x variable. The equation becomes:
y = f(x - a)
Examples
Here are some examples to illustrate how horizontal shifts work:
| Original Equation | Transformation | Shift Description | New Equation |
|---|---|---|---|
| y = x2 | x → (x + 3) | Shift 3 units to the left | y = (x + 3)2 |
| y = | x | x → (x - 2) | |
| y = sin(x) | x → (x + π/2) | Shift π/2 units to the left | y = sin(x + π/2) |
Practical Insights
- Counterintuitive Nature: Note that the direction of the shift might seem counterintuitive. Adding to x shifts the graph to the left, and subtracting from x shifts it to the right.
- Applying to Functions: This method applies to any function f(x). Whether it's a polynomial, trigonometric function, or any other type of function, the same principle applies.
- Combining Transformations: Horizontal shifts can be combined with other transformations (vertical shifts, stretches, reflections) to achieve more complex graph manipulations.
Illustrative Example from Reference
The reference provides an example using y = x2 + 2. To shift this graph 4 places to the right, the equation becomes y = (x-4)2 + 2. This confirms the principle of subtracting from x to shift to the right.
