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Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
The Cartesian Plane is 2 dimensional surface, where positions represent coordinates of the form (x,y). The cartesian plane has
an x-axis: a horizontal line where y=0
a y-axis: a vertical line where x=0.
the origin: where the two axes meet. Both x and y are zero here, so its coordinates are (0,0)
The coordinates (x,y) of a point tell you how far along the x-axis the point is, and how far along the y-axis the point is. By convention, positive x coordinates are to the right of the origin, and positive y-coordinates above the origin. Similarly, negative x and y coordinates are to the left of and below the origin, respectively.
The x and y axes divide the coordinate plane into 4 regions called quadrants. They are often labeled with roman numerals I,II,III and IV.
The distance between two points (x1,y1) and (x2,y2) is given by
The coordinates of the midpoint of two points is
The gradient of the line is a measure of its steepness. It is calculated by measuring the rise (change in y) in the line over a certain run (change in x).
The gradient of the line passing through the points (x1,y1) and (x2,y2) is
A straight line is defined by its gradient and its y-intercept. The gradient-intercept equation of a line is thus:
If we know a point (x1,y1) on a line and the gradient m of the line, we can use the point-gradient form of the line:
A vertical line does not have a well defined gradient, since there is no "run" - the x-values never change.
We cannot write the equation of a vertical line in the form y=⋯. Instead we write
for some constant k.
A horizontal line has gradient m=0. It is therefore in the form
for some constant c.
The equation of a straight line can also be given in the form
This reduces to
In examinations, you may be asked to write the equation of a line in standard form.
Two lines are parallel when they have the same gradient m and they do not intersect:
In this case, the system of equations formed by the two lines has no solutions.
If the lines have the same gradient and they intersect, then they must be the same line.
Suppose we have the straight lines y=3x−2 and y=2−3x. Where do the lines intersect?
Lines intersect when they have a point in common. That is, for some x:
Rearranging gives
If two lines do not intersect, then they must be parallel, since the definition of parallel is two straight lines that never meet.
If two lines are the same (possibly in different forms), then their intersection will all real numbers.
Two lines are perpendicular if they form a right angle with respect to each other. In this case, the rise of one line becomes the run of the other, with a sign change:
We can solve a system of 2 equations and 2 unknowns with different methods.
Rearranging
Substituting this into 3y−3x+1=0:
So x=53, which implies y=−32⋅53+32=154. So the intersection is (53,154).
We can eliminate y from the equations by subtracting the second from the first:
So x=53⇒y=154 and the intersection is again (53,154).
We can use either of these methods to systems of equations with 2 equations and 2 unknowns.
Track your progress across all skills in your objective. Mark your confidence level and identify areas to focus on.
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
Track your progress:
Don't know
Working on it
Confident
📖 = included in formula booklet • 🚫 = not in formula booklet
The Cartesian Plane is 2 dimensional surface, where positions represent coordinates of the form (x,y). The cartesian plane has
an x-axis: a horizontal line where y=0
a y-axis: a vertical line where x=0.
the origin: where the two axes meet. Both x and y are zero here, so its coordinates are (0,0)
The coordinates (x,y) of a point tell you how far along the x-axis the point is, and how far along the y-axis the point is. By convention, positive x coordinates are to the right of the origin, and positive y-coordinates above the origin. Similarly, negative x and y coordinates are to the left of and below the origin, respectively.
The x and y axes divide the coordinate plane into 4 regions called quadrants. They are often labeled with roman numerals I,II,III and IV.
The distance between two points (x1,y1) and (x2,y2) is given by
The coordinates of the midpoint of two points is
The gradient of the line is a measure of its steepness. It is calculated by measuring the rise (change in y) in the line over a certain run (change in x).
The gradient of the line passing through the points (x1,y1) and (x2,y2) is
A straight line is defined by its gradient and its y-intercept. The gradient-intercept equation of a line is thus:
If we know a point (x1,y1) on a line and the gradient m of the line, we can use the point-gradient form of the line:
A vertical line does not have a well defined gradient, since there is no "run" - the x-values never change.
We cannot write the equation of a vertical line in the form y=⋯. Instead we write
for some constant k.
A horizontal line has gradient m=0. It is therefore in the form
for some constant c.
The equation of a straight line can also be given in the form
This reduces to
In examinations, you may be asked to write the equation of a line in standard form.
Two lines are parallel when they have the same gradient m and they do not intersect:
In this case, the system of equations formed by the two lines has no solutions.
If the lines have the same gradient and they intersect, then they must be the same line.
Suppose we have the straight lines y=3x−2 and y=2−3x. Where do the lines intersect?
Lines intersect when they have a point in common. That is, for some x:
Rearranging gives
If two lines do not intersect, then they must be parallel, since the definition of parallel is two straight lines that never meet.
If two lines are the same (possibly in different forms), then their intersection will all real numbers.
Two lines are perpendicular if they form a right angle with respect to each other. In this case, the rise of one line becomes the run of the other, with a sign change:
We can solve a system of 2 equations and 2 unknowns with different methods.
Rearranging
Substituting this into 3y−3x+1=0:
So x=53, which implies y=−32⋅53+32=154. So the intersection is (53,154).
We can eliminate y from the equations by subtracting the second from the first:
So x=53⇒y=154 and the intersection is again (53,154).
We can use either of these methods to systems of equations with 2 equations and 2 unknowns.