Reflections and Vertical Shifts of Trigonometric Graphs
Simple Explanation
A negative sign in front of a trig function (like y=−sinx) reflects its graph over the x-axis, flipping its maximum and minimum. Adding a constant D (like y=sinx+D) shifts the entire graph straight up (or down, if D is negative) by D units.
Why Do We Need It?
These two transformations, combined with amplitude/period/phase shift, complete the full toolkit for describing any transformed trigonometric graph.
See It
Two overlapping wave curves: the basic sine curve, and its reflection over the x-axis shifted up by 1 unit
Formula
Reflections and Vertical Shifts
y = −sin x (reflection over the x-axis); y = sin x + D (vertical shift by D)
A negative sign in front of a trig function reflects its graph over the x-axis; adding a constant D shifts the entire graph up (or down, if D is negative) by D units.
- D
- — the amount the graph is shifted vertically
When to use it: Whenever a trig graph has been flipped or shifted up/down from its basic form.
Worked Example
Describe a reflected and shifted trig graph
Describe how the graph of y=−cos(x)+2 differs from y=cos(x).
Why Does This Work?
Negating every output value of a function flips its graph across the x-axis (a point at height h moves to height −h); adding a constant to every output simply raises (or lowers) the entire graph rigidly by that same amount.
Real-Life Example
Modeling water depth with a non-zero baseline
The depth of water at a tidal dock oscillates around some baseline depth, never actually reaching zero.
A vertical shift (D) accounts for this non-zero baseline depth, while a reflection may be needed depending on whether high or low tide corresponds to the wave's starting behavior.
Practice
Find the maximum value of y=−sin(x)+4.
MediumCommon mistake
Losing track of the order transformations apply in for more complex combined expressions — being explicit about "reflect first, then shift" (matching the order the expression is written) avoids confusion.
Quick Review
- y=−sinx reflects the graph over the x-axis.
- y=sinx+D shifts the graph vertically by D.
- Both can combine with amplitude, period, and phase shift for a fully general transformation.