Coordinate Geometry, Graphs and Circles Flashcards
What do parallel lines have in common?
Gradient
What is of not about the gradients of perpendicular lines?
They multiply to give -1.
What is a perpendicular line called with regards to another?
The normal.
What does it mean if two variables are in direct proportion?
Changing one variable will change the other by the same scale factor so multiplying by any constant will have the same effect on both.
How is “y is directly proportional to x” written?
“y = kx” where k is the constant or constant of proportionality or “y (stretched lower case alpha) x”
What does it mean if two variable are inversely proportional?
Doubling one halves the other, so on.
How is “y is inversely proportional to x” written?
“y = k(1/x)” where k is the constant or constant of proportionality or “y (stretched lower case alpha) 1/x”
What are the three equations that can be used to describe a straight line?
- y-b = m(x-a) for point (a,b) on the line
- y = mx+c
- ax+by+c = 0 where a, b and c are integers
What is the equation for the midpoint of a line segment?
Midpoint of the line (ab) = ((xa+xb)/2 , (ya+yb)/2)
How do you find the length of a line segment?
Pythagoras’ Theorem: length of line (ab)^2 = (xb-xa)^2 + (yb - ya)^2
What are the two general curves for any graph in the form y = kx^n (where n is positive?
- When n is even you get an u/n shape when k is positive/negative.
- When n is odd you get a corner to corner shape with a stationary point in the middle.
What information do you need to sketch a graph and how do you find it?
You need to know where the graph crosses the axes:
- To find where a graph crosses the y-axes, just set x = 0 and find the value of y.
- To find where it crosses the x-axis, factorise the polynomial - it crosses the x-axis when each bracket is set to equal to 0.
If you factorise a polynomial and get a squared bracket what does this mean?
This is a double root, and the graph will only touch the x-axis, not cross it.
If you factorise a polynomial and get a cubed bracket what does this mean?
This is a triple root, which still crosses the x-axis, but flattens out as it does so.
What is a reciprocal function?
Reciprocal functions are those of the form y = k/x^n or y = kx^-n where k is a constant.
What is the general shape of any reciprocal graph when n is even?
An L shape which appears also reflected in the y axis. If k is negative then the whole graph is reflected in the x-axis.
What is the general shape of any reciprocal graph when n is odd?
An L shape which appears also, reflected in the line y = -x. If k is negative the whole thing is reflected in the y-axis.
What is a translation?
Moving the graph of a function either horizontally or vertically. The shape of the graph itself doesn’t change, it just moves.
What kind of translation is y = f(x) + a?
Adding a number to the whole function which translates the graph in the y-direction.
-If a > 0, the graph goes upwards.
-If a < 0, the graph goes downwards.
This can be described by a column vector:
(0)
(a)
What kind of translation is y = f(x + a)?
Writing 'x + a' instead of 'x' translates the graph in the x-direction. -If a > 0, the graph goes to the left. -If a < 0, the graph goes to the right As a column vector, this would be: (-a ) ( 0 )
What are stretches and reflections?
The graph of a functions can be stretched, squashed or reflected by multiplying the whole function or the x’s in the function by a number. The result you get depends on what you multiply and whether the number is positive or negative.
What are the characteristics of stretching a graph like: y = af(x)
Multiplying the whole function by a stretches the graph vertically by a scale factor of a.
-If a > 1 or a < -1, the graph is stretched.
-If a < 1 or a > -1, the graph is squashed.
-If a is negative, the graph is also reflected about the x-axis.
For every point on the graph, the x-coordinate stays the same and the y-coordinate is multiplied by a.
What are the characteristics of stretching a graph like: y = f(ax)?
Writing ‘ax’ instead of ‘x’ stretches the graph horizontally by a scale factor of 1/a.
-If a > 1 or a < -1, the graph is squashed.
-If -1 < a < 1, the graph is stretched.
-If a is negative, the graph is also reflected about the y-axis.
For every point on the graph, the y-coordinate stays the same and the x-coordinate is multiplied by 1/a.
What is the equation of a circle?
(x - a)^2 + (y - b)^2 = r^2, where r is the radius and the centre is (a, b).
How do you put an un-factorised equation of a circle into the correct form?
Complete the square.
What are the important properties of a circle?
- The triangle formed from the ends of a diameter has a right angle.
- The perpendicular from the centre to a chord bisects the chord.
- A tangent to the circle meets a radius at a right angle.
What is are circumcircles?
Circles that pass through all three vertices of a triangle.
