A-level Mathematics/OCR/C1/Coordinate Geometry and Graphs

Co-ordinates are a way of describing position. In two dimensions, positions are given in two perpendicular directions, x and y.

Linear Equations

Straight lines

A straight line has a fixed gradient. The gradient of a line and its y intercept are the two pieces of information that distinguish one line from another line.

Equations of a straight line

The most common form of a straight line is y = mx + c. m is the gradient of the line, and c is the point at which the line intercepts the y-axis. When c is 0, the line passes through the origin (0,0). Other forms of the equation are x = a, used for vertical lines of an infinite gradient and y = b is used for horizontal lines with a zero gradient. Also some equations are commonly written as: px + qy + c = 0. You can also use the equation y-y₁=m(x-x₁)

Gradient of a line

The steepness of a line can be measured by its gradient, which is the change in the y direction divided by the change in the x direction. The letter m is used to denote the gradient. The formula to find a gradient is: $m=\frac{y_2-y_1}{x_2-x_1}$ As a side note $\tan \theta =m$.

Point-Gradient Form

The equation of a line having the co-ordinate $(x_1,y_1)$ and having a gradient of m is: $y-y_1=m(x-x_1)$. Then you simply rearrange the equation into the form y = mx + c.

Parallel lines

A pair of lines are parallel (the symbol is $\Vert$ ). If their gradients are equal, $m_1=m_2$. So in order to find the equation of a parallel line you need the slope of the original line and one set of co-ordinates on the parallel line. Then you use the Point-Gradient formula to find the equation of the parallel line.

Perpendicular lines

A pair of lines are perpendicular (the symbol is $\perp$) if the product of their gradients is $m_1\times m_2=-1$, . So if you need the equation of a line perpendicular to another line, all you need to do is replace the gradient m with the negative reciprocal of m.

So for example if line 1 is y = 2x +3 and you need to find the line perpendicular to this that goes through the point (0,1), then the gradient m = -1/2 (because 2 x -1/2 = -1).

This gives y = -x/2 +c.

Putting the known point (0,1) into this equation gives :

1= -0/2 +c which gives c = 1

so the equation is y = 1- x/2.

Distance between two points

Using the co-ordinates of two points, it is possible to calculate the distance between them using Pythagoras' theorem. The distance d between any two points $A(x_1,y_1)$ and $B(x_2,y_2)$ is given by: $d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$

Mid-point of a line

When the co-ordinate of two points are known, the mid-point is the point halfway between those two points. For any two points A$(x_1,y_1)$ and B$(x_2,y_2)$, the co-ordinates of the mid-point of AB can be found by $(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$.

Intersection of lines

Any two straight lines will meet at a point, as long as they are not parallel. You can find the point of intersection simply by solving the two [[../Equations#Simultaneous equations|equations simultaneously]]. This is also true for curves, although non linear curves may intersect at multiple points or not at all and usually require different methods to solve.

Curves

To sketch a graph of a curve, all you need to know is the general shape of the curve and other important pieces of information such as the x and y intercepts and the points of any maxima and minima.

Curves in the form $y=x^n$

Degree 0 - Constant - $c$ or $k$. In this case y=2	Degree 1 - Linear - $ax+b$ or $mx+c$. In this case y = x.
Degree 2 - Quadratic - $a{x}^2+bx+c$ . In this case y = $x^2$	Degree 3 - Cubic - $a{x}^3+b{x}^2+cx+d$. In this case y = $x^3$

Note: That all the odd powers of $x$ share the same general shape, moving from bottom-left to top-right, and that all the even powers of $x$ share the same "bucket" shaped curve.

Curves in the form $y=\frac{1}{x^n}$

Just like earlier, curves with an even powers of $x$ all have the same general shape, and those with odd powers of $x$ share another general shape.

$y=\frac{1}{x}$	$y=\frac{1}{x^2}$

All curves in this form do not have a value for $x=0$, because $\frac{1}{0}$ is undefined. There are asymptotes on the $y$ axis, where the curve moves towards the y axis increasingly slowly but will never actually touch.

Curves in the form $y=\sqrt{x}$

All curves in this form will not have values for x < 0. They will all have the same general shape.

Intersection of lines and curves

When a line intersects with a curve, it is possible to find the points of intersection by substituting the equation of the line into the equation of the curve. If the line is in the form $y=mx+c$, then you can replace any instances of $y$ with $mx+c$, and then expand the equation out and then [[../Equations|factorise]] the resulting quadratic.

Need information on describing the geometrical relationship between a curve and a straight line

Intersection of curves

The same method can be used as for a line and a curve. However, it will only work in simple cases. When an algebraic method fails, you will need to resort to a graphical or Numerical Method. In the exam, you will only be required to use algebraic methods.

Graphical Rules

In many cases it is easy to obtain a graph from a preexisting graph using these rules.

Reflection

1. $y=-f\left(x\right)$ is a reflection of $y=f\left(x\right)$ through the x axis.
2. $y=f(-x)$ is a reflection of $y=f\left(x\right)$ through the y axis.

Stretching

1. $y=f\left(bx\right)$ is stretched away from the y-axis if $0<b<1$ and stretched towards the y-axis if $b>1$. In both cases the change is by b units.
2. $y=af\left(x\right)$ is stretched towards the x-axis if $0<a<1$ and stretched away from the x-axis if $a>1$. In both cases the change is by a units.

Translations

1. $y=f(x-h)$ is a translation of f(x) by h units to the right.
2. $y=f(x+h)$ is a translation of f(x) by h units to the left.
3. $y=f\left(x\right)+k$ is a translation of f(x) by k units upwards.
4. $y=f\left(x\right)-k$ is a translation of f(x) by k units downwards.

The Circle

A circle is a set of all points in a plane that are a fixed distance r away from a given point called the center. The distance r is the radius of a circle.

The two basic laws of circles are:

$Area=\pi r^2$

$Circumference=2\pi r$

Properties of a circle

A radius always bisects a chord if they are perpendicular to each other.

If any point on the circumference of the circle is connected to the diameter, it forms a right angled triangle.

If a radius is drawn and then a tangent is drawn from that point. Then the radius and tangent line will be perpendicular to each other.

Equation of a Circle

The standard equation of a circle is:

$x^2+y^2=r^2$

This will always give us a circle centered around the origin (0,0). If we want a circle with a center at (h,k) we use the following formula.

${(x-h)}^2+{(y-k)}^2=r^2$

However you can not graph either of these equations with a calculator. You will need to split it into two equations, but the graph will not be perfect, because x is undefined when x = 0 and when $(x-h{)}^2=r^2$: $y=+\sqrt{r^2-{(x-h)}^2}+k$ and $y=-\sqrt{r^2-{(x-h)}^2}+k$.

Here is how ${(x-5)}^2+{(y-8)}^2=2^2$ would look graphed: