Curve sketching
Posted in Maths, Core 1Curve sketching is an important skill you will need to use throughout all of your A level exams.
Graphs of straight lines
You should be able to draw graphs of straight lines from GCSE. Things to remember are that a line with equation y = mx + c has a gradient m and crosses the y-axis at c. The point where it crosses the x axis is given when y = 0.
Quadratic graphs
Graphs of quadratic functions, those with the highest term of x2 have either a U or upside down U shape graph. Which way up the curve is depends on the coefficient of x2. If it's positive the graph will be U shaped else it will be an upside down U:
A quadratic if an equation of ax2 + bx + c crosses the y axis at c just as a straight line.
Minimum and maximum points
A quadratic in the form ax2 + bx + c has a minimum point at if it is a positive graph else it has a maximum at the some point if it is negative.
If a quadratic is completed square format: (x + a)2 + b, then the graph is a minimum or maximum at (-a, b).
Roots
The points where a graph crosses the x-axis are called roots and can be found by factorising the quadratic into the form (ax + b)(cx + d). The x values where the graph cuts the x axis are at -b and -d. Because a quadratic graph is symmetrical the maximum or minimum lines between these two points.
Repeated roots
Some quadratics will only have a single root known as a repeated root. The graph does not cut the x-axis at the point of the root but just touches it:
Cubic graphs
Cubic graphs have a highest power of x3 and take the form ax3 + bx2 + cx + d. The general shape of positive (a > 0) cubic graph is:
Whereas a negative (a < 0) cubic graph takes the form of:
The curve cuts the y-axis at d. Cubic graphs generally have 3 roots or 1 real and 1 repeated root which like quadratic graphs can be found from factorising (see: factorisation to revise factorising polynomials).
Simple transformations
For AS level maths you need to be able to undertake some basic transformation problems such as translation - moving the whole graph up or down and from side to side.
Vertical translation
Take the graph f(x) = x2:
If you were to draw another graph on the same axis, f(x) + a, then the whole graph would move up by a distance of a. If a is negative the graph will move down. The shape of the graph will remain the same however the intersection with the y axis will change from the original to +k.
Horizontal translation
Horizontal translation is slightly more difficult mainly because of the notation which works the opposite way to what you think it would. If you wanted to translate a graph, f(x), a distance b to the right then the equation of the new graph would be f(x - b). If you wanted to move the graph to the left the new equation would be f(x + b).