How do I sketch a function from its equation without using my graphical calculator?
When sketching a graph there are some important points to consider - if you can pinpoint these special places on your graph, you will be able to sketch the whole shape of the graph.
Intercepts - these are the points where your graph crosses the axes. Remember that a graph doesn't need to intercept both axes. You can find the y-intercept(s) by setting x = 0 - this is because if your x value is zero, that puts you in line with the whole of the y-axis. Similarly, you find the x-intercept(s) by setting y = 0. There might be many intercepts and they could be repeating at regular intervals (like in a sine graph), so don't worry if you get more than one answer when you solve your equation.Asymptotes - An asymptote is a value (it could be either an x value or a y value) that you get closer and closer to, but never quite reach. On a graph, they look like a horizontal, vertical, or slanted line that the graph approaches but never touches. As your line will never actually reach the asymptote, we say that it reaches that value at infinity. It's important to recognise asymptotes in your equation. They might look like if your function is a fraction, values where the denominator of your fraction is equal to zero - if the denominator is equal to zero, the value of the fraction tends to infinity so by looking at our definition above, we can recognise this as an asymptotesome trigonometric functions have asymptotes, like the tan function. This is because tan is a fraction of sin/cos, so by using the logic above we see that when cos(x) = 0, you will get an asymptote in your tan graphHow your graph tends to its asymptotes - it's important to know if your graph will approach the asymptote from above or below so you can understand the shape. This is easy to do. If you know that you have an asymptote at x=1 (e.g. when x=1, y tends to infinity), find out what your x value is when y is very large ( e.g. 100000), just by plugging a large number into your equation. If the x value is just below 1, your graph must be tending to 1 from below. If the x value is just above 1, your graph must be tending to 1 from above.
These are general tips for sketching any graph. However you should also know the shapes of common functions (e.g. x2, x3, sin, cos etc.) and you should know how these can be stretched and translated in both the x and y directions
