# How do I horizontally translate a trigonometric graph?

Nov 17, 2014

By changing the "c" in your basic trigonometric equation.

The standard trig equation for sine is $y = a \cdot \sin \left[b \left(x - c \pi\right)\right] + d$. In this, the variable $a$ represents the amplitude. The variable $b$ represents the period ($\frac{2 \pi}{b}$ = period). Now, the variable $c$ represents what is known as the phase shift - more commonly known as a horizontal translation. You shift the graph $c \pi$ units from the original parent function, which in this case is $y = \sin x$. If $c$ is positive, shift the graph to the right $c \pi$ unites. If $c$ is negative, shift the graph to the left $c \pi$ units.

If you're wondering, $d$ represents the vertical translation.

I hope this helps, and I'f strongly suggest going to google and typing in functions like $y = \sin \left(x - 2 \pi\right)$ and comparing them to the parent function, $y = \sin x$.

Nov 17, 2014

Generally speaking, if you have a function $y = f \left(x\right)$ and know its graph, the function $y = f \left(x - c\right)$ has a graph that is similar to the one of $y = f \left(x\right)$ but shifted by $c$ to the right for $c > 0$ or to the left for $c < 0$.

Continuing the graph transformation, the graph of $y = f \left(x\right) + d$ is similar to the graph of $y = f \left(x\right)$ but shifted by $d$ up for $d > 0$ or down for $d < 0$.

Next transformation is related to a graph of a function $y = a \cdot f \left(x\right)$. The graph of this function can be obtained from a graph of $y = f \left(x\right)$ by stretching (if $| a | > 1$) or squeezing (if $| a | < 1$) it by a factor $a$ vertically. That is, point $\left(x , y\right)$ on a graph of $y = f \left(x\right)$ is transferred into $\left(x , a \cdot y\right)$ on a graph of $y = a \cdot f \left(x\right)$. This includes reflection relative to the X-axis for $a < 0$.

Finally, the graph of a function $y = f \left(b \cdot x\right)$ can be obtained from the graph of $y = f \left(x\right)$ by horizontal squeezing (if $| b | > 1$) or stretching (if $| b | < 1$) it by a factor of $b$. That is, point $\left(x , y\right)$ on a graph of $y = f \left(x\right)$ is transferred into $\left(\frac{x}{b} , y\right)$ on a graph of $y = f \left(b \cdot x\right)$. This includes reflection relative to the Y-axis for $b < 0$.

You can find more detailed explanation of these manipulations with graphs in a lecture on Unizor following the menu items Algebra - Graphs - Transformation.