Sketching Curves
Quadratics
Method for working out the points on Quadratics:
1. Find out where the axis are intercepted
a) Find out values of x by factorisation
b) Find out y intercept by using c in the equation ax^2 + bx + c = y
2. Find either the minimum or maximum point by completing the square
1. Find out where the axis are intercepted
a) Find out values of x by factorisation
b) Find out y intercept by using c in the equation ax^2 + bx + c = y
2. Find either the minimum or maximum point by completing the square
Cubics
Method for working out the points of a Cubic:
1. Find out where the axis are intercepted
a) Find out x values by factorisation if need to
b) Find out y intercept by multiplying the integers in the brackets once factorised
2. ± ∞ tendancies.
When all integers are multiplied together if they give a positive number they give a positive cubic graph, when a negative they give a negative cubic graph
y = (+)(+)(+) giving a positive cubic
y = (-)(-)(-) giving a negative cubic
1. Find out where the axis are intercepted
a) Find out x values by factorisation if need to
b) Find out y intercept by multiplying the integers in the brackets once factorised
2. ± ∞ tendancies.
When all integers are multiplied together if they give a positive number they give a positive cubic graph, when a negative they give a negative cubic graph
y = (+)(+)(+) giving a positive cubic
y = (-)(-)(-) giving a negative cubic
Reciprocals
The larger the numerator the closer the line appears towards the axis. No way of plotting any points.
Transformations
Transforming 'y = f(x)' where f(x) can be a quadratic or cubic
1) y = -f(x)
2) y = f(-x)
3) y = f(x+a)
4) y = f(x) + a
p d d 5) y = f(ax)
d d d d d d d 6) y = af(x)
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Mirrors on the x axis - this means swap the signs for the y coordinates
Mirrors on the y axis - this means swap the signs for the x coordinates
When:
a < 0 then move to the right 'a' amount of times a > 0 then move to the left 'a' amount of times E.g. if a=2 then move 2 to the left When:
a < 0 then move down 'a' amount of times a > 0 then move up 'a' amount of times E.g. if a=2 then move up 2 When:
a > 1 then x values are divided by 'a' - it steepens a = 0 then it is a flat line a < 1 then x values are divided by 'a' - it spreads E.g. if a=3 then all x values are divided by 3 When:
a > 1 then y values are multiplied by 'a' - it becomes thinner a < 1 then y values are multiplied by 'a' - it becomes wider E.g. if a=3 then all y values are multiplied by 3 |
Briefly: When 'a' is within the brackets it affects the x axis and does the opposite action. When outside the brackets it affects the y axis and does what is expected.