# How do you graph y=sec(x+4)?

Jul 14, 2015

Start with a graph of $y = \cos \left(x\right)$, then invert it getting $y = \frac{1}{\cos} \left(x\right) = \sec \left(x\right)$, then shift the graph to the left by $4$.

#### Explanation:

Since $\sec \left(x\right)$ is, by definition, $\frac{1}{\cos} \left(x\right)$, and $y = \cos \left(x\right)$ is a very familiar function with a very well known graph, let's start with a graph of $y = \cos \left(x\right)$.

graph{cos(x) [-10, 10, -5, 5]}

Next step is to transform this graph into $y = \frac{1}{\cos} \left(x\right) = \sec \left(x\right)$. To accomplish this, we notice that everywhere, where $\cos \left(x\right) = 0$, $\frac{1}{\cos} \left(x\right)$ has vertical asymptote. The sign of $\cos \left(x\right)$ and $\frac{1}{\cos} \left(x\right)$ is the same, symmetry considerations and periodicity are the same as well. Also, when $\cos \left(x\right)$ increases in absolute value from $0$ to $1$, $\frac{1}{\cos} \left(x\right)$ decreases in absolute value from $\infty$ to $1$.

So, the graph of $y = \frac{1}{\cos} \left(x\right) = \sec \left(x\right)$ looks like this:

graph{1/cos(x) [-10, 10, -5, 5]}

Finally, if you have a graph of function $y = f \left(x\right)$, you can easily construct a graph of function $y = f \left(x + A\right)$. You just shift the graph by $A$ units to the left (for positive $A$) or shift it to the right by $- A$ units (for negative $A$).

So, here is a graph of $y = \frac{1}{\cos} \left(x + 4\right) = \sec \left(x + 4\right)$:

graph{1/cos(x+4) [-10, 10, -5, 5]}

A detailed explanation of different techniques used in creating graphs of algebraic and trigonometric functions you can find at Unizor by following the link Algebra - Graphs.