- Gradient Intercept Form of an equation of a line helps plot the graph:
- If we are to sketch a curve, we find the gradient by differentiating the equation to find the derivative.
- An asymptote of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity.
- Revise the rules for graphical transformations.
Firstly, let’s quickly recall from the article “coordinate geometry” , the equation of a straight line.
We studied 3 different forms in which an equation of a line can be written as:
1. Gradient Intercept Form
Equation of a line with gradients m and y intercept c is:
2. Gradient Point Form
If a line with gradient m, passes through a point (x1, y1), then its equation is:
3. Two Point Form
Equation of a straight line passing through two points (x1 ,y1) and (x2, y2) is:
If we are given the gradient intercept form, we can find the gradient of the line and also the point c which is the point at which line intersects the y axis. If we are given these two values, we can easily draw a straight line on the graph by substituting different values of x in the equation and finding the corresponding values of y.
Q. Show the line on a graph.
We know gradient m = 2 &
y intercept c = 5
Let’s take 5 random values of x and find the corresponding values of y.
Now sketch the line
Moving on, it gets a little more complicated when we have to sketch curves. If we have an equation of a curve, we use the same method as above, following the steps below:
- First find the derivative of the equation which is the gradient of the curve.
- Substitute values of x in the equation and find the corresponding values of y.
- Find maximum or minimum points through double derivation.
Another important point to note is if an asymptote occurs. An asymptote of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity.
For example if we have a function and when x = 2 then f(x) becomes which is equal to infinity and thus we get an asymptote at x = 2.
A very popular graph in which asymptotes exist is the graph of tan x.
Here we can observe an asymptote at and .
Recalling from the article “Functions”, the rules listed below are very important and must be remembered when sketching curves of different functions.
Any small change to the equation of a function could lead to a big change in its graph.
Suppose we have a function , some rules to remember when transforming graphs are:
- The graph of where c is a constant, has the same shape of the function f(x) but is moved c units higher.
- The graph of where c is a constant, has the same shape of the function f(x) but is moved c units lower.
- The graph of , where k is a constant, has the same shape of the function but is moved k units to the left.
- The graph of , where k is a constant, has the same shape of the function but is moved k units to the right.
- The graph of is simply a reflection of f(x) in the y axis.
- The graph of is a reflection of f(x) in the x axis.
- The graph of is stretched by a scale factor a in the y axis.
- The graph of is stretched by a scale factor in the x axis.