Curve Sketching | Examples & Summary | A Level Maths Revision Notes

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1 Summary

1.1 The 3 different forms in which an equation of a line can be written as:

1.1.1 Example #1

1.2 Graphical Transformations

1.2.1 Reference

Summary

Gradient Intercept Form of an equation of a line helps plot the graph: $y\quad =\quad mx\quad +\quad c$
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:

$y\quad =\quad mx\quad +\quad c$

2. Gradient Point Form

If a line with gradient m, passes through a point (x1, y1), then its equation is:

$y\quad -\quad { y }_{ 1 }\quad =\quad m(x\quad -\quad { x }_{ 1 })$

3. Two Point Form

Equation of a straight line passing through two points (x1 ,y1) and (x2, y2) is:

$\frac { y\quad -\quad { y }_{ 1 } }{ { y }_{ 2 }\quad -\quad { y }_{ 1 } } \quad =\quad \frac { x\quad -\quad { x }_{ 1 } }{ { x }_{ 2 }\quad -\quad { x }_{ 1 } }$

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 $y\quad =\quad 2x\quad +\quad 5$ 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 $x\quad =\quad -2$

$y\quad =\quad 2(-2)\quad +\quad 5\quad =\quad 1$

When $x\quad =\quad -1$

$y\quad =\quad 2(-1)\quad +\quad 5\quad =\quad 3$

When $x\quad =\quad 0$

$y\quad =\quad 2(0)\quad +\quad 5\quad =\quad 5$

When $x\quad =\quad 1$

$y\quad =\quad 2(1)\quad +\quad 5\quad =\quad 7$

When $x\quad =\quad 2$

$y\quad =\quad 2(2)\quad +\quad 5\quad =\quad 9$

Now sketch the line $y\quad =\quad 2x\quad +\quad 5$

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 $f\left( x \right) \quad =\quad \frac { 1 }{ x\quad -\quad 2 }$ and when x = 2 then f(x) becomes $\frac { 1 }{ 0 }$ 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 $x\quad =\quad \frac { \pi }{ 2 }$ and $x\quad =\quad \frac { 3\pi }{ 2 }$ .

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 $y\quad =\quad f\left( x \right)$ , some rules to remember when transforming graphs are:

The graph of $y\quad =\quad f\left( x \right) \quad +\quad c$ where c is a constant, has the same shape of the function f(x) but is moved c units higher.
The graph of $y\quad =\quad f\left( x \right) \quad -\quad c$ where c is a constant, has the same shape of the function f(x) but is moved c units lower.
The graph of $y\quad =\quad f\left( x\quad +\quad k \right)$ , where k is a constant, has the same shape of the function but is moved k units to the left.
The graph of $y\quad =\quad f\left( x\quad -\quad k \right)$ , where k is a constant, has the same shape of the function but is moved k units to the right.
The graph of $y\quad =\quad f\left( -x \right)$ is simply a reflection of f(x) in the y axis.
The graph of $y\quad =\quad -f\left( x \right)$ is a reflection of f(x) in the x axis.
The graph of $y\quad =\quad af\left( x \right)$ is stretched by a scale factor a in the y axis.
The graph of $y\quad =\quad f\left( ax \right)$ is stretched by a scale factor $\frac { 1 }{ a }$ in the x axis.