Graphing and Manipulating Equations

Graphing Linear Equations

Graphing linear equations only require two point: the y intercept and the x intercept. For example, say you have the general line y=mx+c. To find the y intercept, set x equal to 0, so you get y=c, which is why its called the y intercept. To find the x intercept, set y equal to 0 to get mx+c=0 => mx=-c => x=-c/m. Easy

Graphing Quadratic Equations

When asked to graph a quadratic, there are some things you have to look out for:

There is always a y-intercept, given by the '+c' term
When checking to see if it intercepts the x-axis, check the discriminant(b²-4ac) too see if it actually intercepts. Look back if you cannot remember.
If it does intersect the x-axis, then it can be solved.

Graphing Cubic Equations

Graphing cubic equations is similar to how quadratic graphs are graphed, however it always intersects the x-axis at least once. The y-intercept comes from the '+d' at the end, and to find the x-intercept(s) the cubic must be factorised and set equal to 0

Intersection of Graphs

If you get asked in a question to find where two graphs intersect, it's just as simple as finding at what x-value(s) the y value(s) is/are equal, and then finding the y-values from the x-value, and sometimes showing it on a graph.

Effects of Translations on Graphs

Quadratics

A translation of a across and b up means that the graph is being moved a units in the positive x-direction and b units in the positive y-direction. Now, showing this on a graph is easy as you can move the vertex and therefore the whole graph. Doing it to the equation, however is a bit more difficult. Firstly, you'll want to turn the graph into the completed square form. Lets keep it easy to start with and do the translation of a across and b up to the y=x² graph, as it is already in its completed sqaure form. For the b up part, that is easy as it's just +b, so now it's y=x²+b. For the move a across part however, we want to take a away from x, so we get y=(x-a)²+b. Why are we taking a away? That is because it affects the x part of the equation, and so the opposite of what you think would happen happens. The b is added as it affects only the y. For example, if the graph y=x²+4x+3 is translated 3 down and 2 to the left, then firstly it is turned into its completed square form, y=(x+2)²-1. So by taking into account the translation, it is now y=(x+4)²-4.

Circles

Firstly, circles that have the origin at (0,0) have the equation x²+y²=r², where r is the circle's radius. Its x-intercept is (r,0) and its y-intercept us (0,r). When a circle is translated by a across and b up, it then gets the equation (x-a)²+(y-b)²=r². Now, you may be asking, why are they both taking away, and not adding. Well then, take a random point on the circle (x,y). The horizontal distance to the point from the origin (a,b) is x-a, and the vertical distance is y-b. Using Pythagoras, r² = (x-a)²+(y-b)², hence the general equation.

Summary

When graphing a linear equation (y=mx+c), the x-intercept is -c/m and the y-intercept is c.
When graphing a quadratic, the x-intercept(s) either:

Don't exist (see the page on the discriminant)
Or are findable by solving through one of the 3 methods

The y intercept of a quadratic is c
When graphing a cubic, usually you are asked to find it's factors earlier on. If so, set the factors equal to 0 and there is/are the x-intercept(s). the y intercept is d
Graphs intersect where their y values are equal (basically, just set them equal to each other).
When translating a y=x² quadratic by a across and b up, it becomes y=(x-a)²+b graph.
When translating a x²+y²=r² circle by a across and b up, it becomes (x-a)²+(y-b)²=r².