Vertical Changes to a Sinusoid
A sine or cosine graph repeats the same pattern of rises and falls. Two simple changes alter where that pattern sits vertically and which way its peaks and troughs face: adding a constant shifts the graph, while multiplying by a negative number reflects it.
These changes can be read directly from the function. They determine the graph’s midline, its highest and lowest values, and whether its usual orientation has been reversed. The horizontal placement and period may also be encoded in the function, but here the focus is on the vertical features.
Midline, Vertical Shift, and Reflection
Write a transformed sine or cosine function in the form
where \(A\ne 0\), \(B>0\), and \(C,D\) are real numbers. The parameters \(B\) and \(C\) describe horizontal behavior. The parameters \(A\) and \(D\) control the vertical behavior. In particular, the horizontal line \(y=D\) is the graph’s midline.
Adding \(D\) moves every output by \(D\): positive \(D\) shifts the graph upward and negative \(D\) shifts it downward. The sign of \(A\) has a different effect. A negative sign reverses the vertical displacement from the midline: points above the midline move below it, and points below it move above it.
It is important to identify the line of reflection correctly. When \(D=0\), reflecting across the midline means reflecting across the \(x\)-axis. When \(D\ne 0\), the reflection is across \(y=D\), not across the \(x\)-axis.
Range and Extreme Values
For every real input \(u\), the sine and cosine values lie between \(-1\) and \(1\), inclusive. Multiplication by \(A\) scales these values, and addition of \(D\) shifts them. This gives the range and extrema for either function form.
The absolute value is essential: the size of the vertical displacement from the midline is \(|A|\), regardless of the sign of \(A\). A negative coefficient changes which phase points produce the maximum and minimum, but it does not change the range.
Why the Range Formula Works
Here is a direct justification. Set \(u=B(x-C)\). As \(x\) ranges over all real numbers and \(B>0\), \(u\) also ranges over all real numbers. For either sine or cosine, write the trigonometric value as \(t\), where \(-1\le t\le 1\). Then \(-|A|\le At\le |A|\), whether \(A\) is positive or negative. Adding \(D\) to all three parts gives
Thus every output is between \(D-|A|\) and \(D+|A|\). The endpoints are attained because sine and cosine each attain both \(-1\) and \(1\). So these bounds are the actual minimum and maximum, not just estimates.
The same reasoning explains the reflection. Let \(g(x)=|A|\sin(B(x-C))+D\), and let \(f(x)=A\sin(B(x-C))+D\) with \(A<0\). Since \(A=-|A|\),
A point with vertical coordinate \(g(x)\) is therefore moved to the coordinate \(2D-g(x)\), which is the same distance on the opposite side of \(D\). This is precisely reflection across the horizontal line \(y=D\). The proof for cosine is identical after replacing sine with cosine.
A Reliable Sketching Process
To sketch a sinusoid when its equation is given, first mark the midline and the two extreme values. Then use the sine or cosine key points in the angle input, taking care to multiply their vertical displacements from the midline by \(A\). This makes the effect of a negative coefficient visible without confusing it with a shift.
Identify \(A\) and \(D\). The midline is \(y=D\).
The minimum is \(D-|A|\), and the maximum is \(D+|A|\).
Evaluate the sine or cosine at its usual key angles, then apply multiplication by \(A\) and addition of \(D\).
A negative \(A\) exchanges peaks and troughs relative to the midline. Confirm that the plotted values stay within the range.
Worked Examples
Worked Example: Shifting a Sine Graph Upward
Find the midline, range, and key values of \(y=2\sin x+3\) over one period, from \(x=0\) to \(x=2\pi\).
Here \(A=2\), \(B=1\), \(C=0\), and \(D=3\). The midline is \(y=3\). The vertical distance from the midline is \(|A|=2\), so the range is
Use the five standard sine key inputs \(0,\ \frac{\pi}{2},\ \pi,\ \frac{3\pi}{2},\ 2\pi\). At these inputs, sine takes the values \(0,1,0,-1,0\). Apply \(y=2\sin x+3\) at each one:
The plotted values rise from the midline to \(5\), return to the midline, descend to \(1\), and return to the midline. They agree with the predicted range \([1,5]\) and complete one sine cycle. As a check, the maximum listed value is \(5\), the minimum is \(1\), and their average is \(\frac{5+1}{2}=3\), the stated midline.
Worked Example: A Reflected and Shifted Cosine Graph
For \(y=-3\cos\bigl(2(x-\frac{\pi}{4})\bigr)-1\), find the midline, range, and five key points over one period. Explain the reflection.
The vertical parameters are \(A=-3\) and \(D=-1\). Therefore the midline is \(y=-1\), and the range is
The angle input is \(2(x-\frac{\pi}{4})\). One full cosine cycle occurs as this input goes from \(0\) to \(2\pi\), corresponding to \(x\) from \(\frac{\pi}{4}\) to \(\frac{5\pi}{4}\). The five key inputs are \(0,\frac{\pi}{2},\pi,\frac{3\pi}{2},2\pi\). Solving \(2(x-\frac{\pi}{4})\) equal to these values gives \(x=\frac{\pi}{4},\frac{\pi}{2},\frac{3\pi}{4},\pi,\frac{5\pi}{4}\).
The corresponding cosine values are \(1,0,-1,0,1\). Substitution gives
The negative coefficient reflects the cosine pattern across \(y=-1\): its starting maximum becomes a minimum, and its minimum becomes a maximum. Check that the listed extremes are \(-4\) and \(2\), exactly the endpoints of the predicted range, and that their midpoint is \(\frac{-4+2}{2}=-1\).
Worked Example: A Reflected Sine Graph with a Downward Shift
Analyze \(y=-\frac{1}{2}\sin(3x)+2\). Give its midline, range, orientation at \(x=0\), and key values over the interval from \(0\) to \(\frac{2\pi}{3}\).
Here \(A=-\frac{1}{2}\), \(B=3\), and \(D=2\). The midline is \(y=2\), and the vertical distance from it is \(\frac{1}{2}\). Thus the range is
Over the stated interval, \(3x\) runs from \(0\) to \(2\pi\), so it covers one sine cycle. Use inputs \(3x=0,\frac{\pi}{2},\pi,\frac{3\pi}{2},2\pi\), which give \(x=0,\frac{\pi}{6},\frac{\pi}{3},\frac{\pi}{2},\frac{2\pi}{3}\). Substituting the sine values \(0,1,0,-1,0\) yields
The graph starts at the midline and moves downward to a minimum, because the positive sine pattern is reflected across \(y=2\). The calculated minimum and maximum, \(\frac{3}{2}\) and \(\frac{5}{2}\), match the range. Their average is \(\frac{\frac{3}{2}+\frac{5}{2}}{2}=2\), confirming the midline.
Common Mistake: Treating Every Negative Coefficient as a Shift Down
A frequent error is to interpret a negative \(A\) as though it lowered the whole graph. For example, for \(y=-2\cos x+3\), one might incorrectly say that the graph has midline \(y=1\), reasoning that the negative coefficient subtracts \(2\) from \(3\). That is not what the expression means: the cosine value changes with \(x\), so the negative coefficient reverses the cosine pattern rather than setting a new constant midline.
The correct calculation is \(D=3\), so the midline is \(y=3\). Since \(|A|=2\), the range is \([3-2,3+2]=[1,5]\). At \(x=0\), \(y=-2(1)+3=1\), a minimum; at \(x=\pi\), \(y=-2(-1)+3=5\), a maximum. The two values sit equally far below and above \(3\), confirming that the reflection is across \(y=3\).
Why These Vertical Features Matter
The midline and range give a quick way to test whether a sketch is plausible. A graph that claims to represent \(A\sin(B(x-C))+D\) must oscillate around \(y=D\) and remain between \(D-|A|\) and \(D+|A|\). A negative \(A\) changes the order of peaks and troughs, not these bounds.
When comparing two sine or cosine equations, first separate vertical changes from horizontal ones. The constant outside the trigonometric function gives the midline; the coefficient multiplying the trigonometric function gives the vertical scale and, through its sign, whether the pattern is reflected. Keeping those roles distinct prevents common graphing errors.
Check Your Understanding
Answer these questions using the relationship between the coefficient, the midline, and the range.
- For \(y=4\sin x-2\), what are the midline and range?
- For \(y=-\cos x+1\), across which horizontal line is the cosine pattern reflected, and what is its range?
- Why does a negative coefficient not necessarily shift the graph below the \(x\)-axis?
- For \(y=-2\sin x+3\), what is the value at \(x=\frac{\pi}{2}\), and is it a maximum or minimum?
- For any nonzero \(A\), explain why the midpoint of the maximum and minimum values is \(D\).
