I've found that quite a few people (dcc included) have problems sketching trig graphs fast. Well do I have the solution for you! I was taught this at school by my teacher (who will remain nameless for privacy concerns) and it has never failed me. So without further ado:
TRINON'S GUIDE TO SKETCHING TRIG GRAPHS = -2sin(3(x-\frac{\pi}{4})), x \in [-\pi,\pi])
Before even attempting to sketch a graph, you want to write down the key components of the graph.
1. 
2. 
3. 
4. 
5. 
6. 
Now we can start to draw the graph.1. Put in dotted lines where the central axis is and where the top value will be and the bottom value will be. To find the top and bottom lines, we simply plus/minus the amplitude from the central axis.
2. Now we need to put in the scale of the x-axis. According to the horizontal translation and the scale factor, we are going to put in a scale according to the lowest common denominator of the two. In this case, the lowest common denominator is
. Noting the domain of the function, mark in the points on the x-axis.
3. Next we are going to put in the marks of the scale factor. The scale factor is a quarter of the period and lies on the central, top and bottom axis. This will help us later sketch the graph. Carefully mark in a small cross where each minimum, maximum and central axis intercept occurs.
4.Now we need to find the axis intercepts.
Y-Axis:
 = -2sin(\frac{-3\pi}{4}) = \sqrt{2}, \therefore (0, \sqrt{2}))
Find the domain for which we will solve for x:
 \leq \frac{9\pi}{4})
X-Axis:
))
 = -3\pi, -2\pi, -\pi, 0, \pi, 2\pi)
5. Next we are finding the endpoints.
 = f(-\pi) = -2sin(\frac{9\pi}{4}) = -\sqrt{2})
 and (\pi, -\sqrt{2}))
6. Now we can mark in the end points and sketch the graph. Be mindful of the intercepts.
Sketching a tan graphThe steps are much the same as sketching a sin or cos graph. The only differences are the period and the introduction of asymptotes.
For a standard
})
graph the period is
. The formula for finding the period is now
.
To find the asymptotes, you first need to find the center of the tan graph on the x-axis. This is done by noting the translations. Lets say have the following equation:
 = \tan{(n(x-h))} + k)
The center of the tan graph will be at
)
. Now to find the asymptotes, you simply mark in half of the period (in this case
) from either side of the center point, then continue every period until the end of the domain.
The center of each tan curve will be directly in between each asymptote. It is now possible to mark in where the other curve centers are, and draw a rough line for what the graph will look like. When doing this, I like to use dotted lines to signify that it isn't actually the curve, but what it may look like.
Last but not least, you let the equation equal 0, and find the x-axis intercepts. This part is identical to finding the x-axis intercepts for the sin and cos graph.
I've included an example below that I did by hand:
And we're done!There you have it. I hope this guide has been helpful. Do realise that once you get the hang of this technique, you can start to leave steps out. For example, on this graph I didn't even bother to find the x-axis intercepts because I could see that they would fall on the scale points that I had already drawn.
Home
Help
Search
Login
