How Do You Graph a Sine Function Increased on One Side?

There is no single graph described by “a sine function increased on one side.” You need to say what “increased” means and where the change begins. It could mean a larger amplitude, a vertical shift, or adding a term that raises the values on one side of a boundary.

A common interpretation is to give the sine wave a larger amplitude for $x \ge 0$. The example below shows how to define and sketch that piecewise function, and how it differs from simply raising the graph.

Example: double the amplitude for $x \ge 0$

Define

$$f(x)=\begin{cases} \sin x, & x<0,\\ 2\sin x, & x\ge 0. \end{cases}$$

For negative inputs, use the usual sine graph. For nonnegative inputs, multiply its values by 2. This stretches the right-hand part vertically: its peaks are twice as high and its troughs twice as deep.

To sketch it, mark the usual sine landmarks on each side of zero. For example:

$x$	Left rule, $\sin x$	Right rule, $2\sin x$
$-\pi/2$	$-1$	—
$0$	Boundary value	$0$
$\pi/2$	—	$2$
$\pi$	—	$0$
$3\pi/2$	—	$-2$

The function is continuous at zero: the left-hand values approach $\sin 0=0$, and the right-hand rule also gives $2\sin 0=0$. But the slope changes there. The derivative from the left is $\cos 0=1$, while from the right it is $2\cos 0=2$. So the joined graph has a corner at the origin rather than a smooth tangent.

This example does not make every value on the right larger than the original sine value. When $\sin x$ is negative, doubling it makes it smaller—for instance, $-2<-1$. Amplitude increase means greater distance from the horizontal midline, not that every function value increases.

If you mean “raise” the right side

To make the right-hand values higher by a fixed amount $c>0$, one possible rule is

$$g(x)=\begin{cases} \sin x, & x<0,\\ \sin x+c, & x\ge 0. \end{cases}$$

This shifts the right-hand sine wave up by $c$. At zero, the left-hand limit is $0$ and the defined right-hand value is $c$, so the graph jumps. If you want the graph to join continuously, use an added term that is zero at the boundary and grows afterward. For example,

$$h(x)=\begin{cases} \sin x, & x<0,\\ \sin x+c(1-e^{-x}), & x\ge 0. \end{cases}$$

Here $c(1-e^{-x})$ is zero at $x=0$ and positive for $x>0$. The two pieces meet at the origin, and the right-hand values are raised increasingly toward an offset of $c$. This changes the shape of the right side; it is not just a constant vertical translation.

A general way to build the graph

First choose the boundary, often $x=0$. Then write one rule on each side. For an amplitude change, use $A\sin x$ on the modified side, where $A$ is the new amplitude. For a vertical shift, use $\sin x+c$. For a gradual increase, add a function that starts at zero at the boundary and grows on the chosen side.

Finally, check the boundary values. A piecewise graph is continuous where the pieces meet if their limiting values agree with the function’s value there. Matching values does not guarantee matching slopes, as the amplitude example demonstrates.

Common mistakes

• Confusing amplitude with a vertical shift: $2\sin x$ changes the distance from the midline; $\sin x+1$ moves the midline up by 1.
• Assuming a larger amplitude means larger values: Negative troughs become more negative when the amplitude increases.
• Ignoring the boundary: Check the limits from both sides to see whether the graph has a jump or a corner.
• Expecting the modified graph to stay periodic: Changing the rule only on one half of the real line generally destroys the original sine function’s global periodicity.

If a particular graph is required, specify the formula, the side being changed, and whether you want the result to be continuous or smooth at the boundary. Those choices determine the graph.