You need to understand, sketch and use graphs of the following functions:
[ul]liCubic functions[/li]liQuartic functions[/li]liReciprocal functions[/li]liModulus of linear functions[/li]/ul
uCubic functions/u
Cubic functions have the general form (ax3+bx2+cx+d), and ones which you have to sketch can usually be factorised into the form ((x+a)(x+b)(x+c)). They can have up to three roots.
Tip: Try to use the topic=2/18factor theorem/topic to find factors quickly.
row
[col]pWhen (a>0), the graph starts at the bottom left, and finishes at the top right.[/p]mtaimg/images/topics/2/2-14-1.png[/mtaimg]/col
[col]pWhen (a<0), the graph starts at the top left, and finishes at the bottom right.[/p]mtaimg/images/topics/2/2-14-2.png[/mtaimg]/col
/row
uQuartic functions/u
Quartic functions have the general form (ax4+bx3+cx2+dx+e), and ones which you have to sketch can usually be factorised into the form ((x+a)(x+b)(x+c)(x+d)). They can have up to four roots.
Tip: Try to use the topic=2/18factor theorem/topic to find factors quickly.
row
[col]pWhen (a>0), the graph starts at the top left, and finishes at the top right.[/p]mtaimg/images/topics/2/2-14-3.png[/mtaimg]/col
[col]pWhen (a<0), the graph starts at the bottom left, and finishes at the bottom right.[/p]mtaimg/images/topics/2/2-14-4.png[/mtaimg]/col
/row
uReciprocal functions/u
Reciprocal functions have the general form (\dfrac{a}{xn}). The shape depends on whether (n) is odd or even, and whether (a) is positive or negative.
bAsymptotes/b (lines which the graph approaches but never touches) are shown as red dashed lines.
[ul]liHorizontal asymptotes are calculated by considering what happens to the graph for very small and large values of (x).[/li]liVertical asymptotes are calculated by considering what happens to the graph for very small and large values of (y).[/li]/ul
[row][col]pWhen (a>0) and (n) is odd, e.g. (y=\dfrac{1}{x})[/p]mtaimg/images/topics/2/2-14-5.png[/mtaimg]/col
[col]pWhen (a<0) and (n) is odd, e.g. (y=\dfrac{-1}{x})[/p]mtaimg/images/topics/2/2-14-6.png[/mtaimg][/col]/row
[row][col]pWhen (a>0) and (n) is even, e.g. (y=\dfrac{1}{x2})[/p]mtaimg/images/topics/2/2-14-7.png[/mtaimg]/col
[col]pWhen (a<0) and (n) is even, e.g. (y=\dfrac{-1}{x2})[/p]mtaimg/images/topics/2/2-14-8.png[/mtaimg][/col]/row
uModulus of linear functions/u
The bmodulus/b of a function means "turning any negative numbers into positive numbers" for that function. The modulus of a linear function is therefore V-shaped. The negative section of the linear function is reflected about the (x)-axis.
The linear function is represented by the purple dashed line, and its modulus is represented by the solid red line.
mtaimg/images/topics/2/2-14-9.png/mtaimg
[b]uIntersections of graphs[/u]/b
The (x)-coordinate(s) at the point(s) of intersection of the graphs (y=f(x)) and (y=g(x)) are the solution(s) of the equation (f(x)=g(x)).
For example, in the diagram below, there are three solutions to the equation (f(x)=g(x)), because there are three intersections.
mtaimg/images/topics/2/2-14-10.png/mtaimg
bAsymptotes/b are lines which the graph approaches but never touches.