Graphs and Curve Sketching

You need to be able to plot straight line graphs and sketch curves for quadratic, cubic and trigonometric graphs. You  need to understand the effects of transformations of graphs. You need to be able to use graphs to find graphical solutions to equations.

Transformations of Graphs

Transformations such as translations and stretches can be applied to alter the position and/or size of a graph.

 

Original

graph

 New

Graph

 Transformation Notes
y = f(x) y = f(x )+ a Translation If a is positive, curve moves up a units.

If a is negative, curve moves down a units
y = f(x) y = f(x + a) Translation If a is positive, curve moves left a units.

If a is negative, curve moves right a units
y= f(x) y = af(x) Stretch Stretch from the x axis, parallel to the y axis, scale factor a.

The y coordinates on the graph of y = f(x) are multiplied by a.
y = f(x) y = f(ax) Stretch Stretch from the y axis, parallel to the x axis, scale factor 1/a.

The x coordinates on the graph of y = f(x) are divided by a.
y = f(x) y = – f(x) Reflection Reflection in the x axis.

The y coordinates on the graph of y = f(x) change signs.
y = f(x) y =  f(- x) Reflection Reflection in the y axis.

The x coordinates on the graph of y = f(x) change signs.

 

As with translations of single points or shapes, vector notation can be used to describe the translations above.

 Curve sketching

You should be able to:

§        Recognise whether an equation represents a curve or a straight line

§        Sketch simple quadratic curves of the form y = x2 + 2, y = x2 – 5 or y = x2

§        Sketch simple cubic graphs of the form y = x3, y = x3 + 1

§        Recognise the basic trigonometric curves y = sin x, y = cos x and y = tan x

§        Sketch the trigonometric curves y = sin x, y = cos x and y = tan x

§        Sketch trigonometric curves such as y = sin 2x, y = 3 cos x

§        Plot a straight line and sketch a curve on the same axes, and use them to find (approximate) intersection points

§        Be able to estimate the solution to an equation by plotting  suitable graph(s)