**Summary**

- 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.

### The 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*.

#### Example #1

Q. Show the line on a graph.

*Solution:*

We know gradient *m = 2* &

*y* intercept *c = 5*

Let’s take 5 random values of *x* and find the corresponding values of *y*.

x | -2 | -1 | 0 | 1 | 2 |

y | 1 | 3 | 5 | 7 | 9 |

When

When

When

When

When

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.

### Graphical Transformations

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
*x*axis.