Solving Quadratic Equations
Example
Figure 1 shows the graph of a quadratic function. Find the formula for this quadratic
function and express the formula in vertex form and in standard form.
Figure 1: Find the formula of this quadratic function.
Solution
We will find the formula in vertex form (this is relatively easy as the x and ycoordinates
of the vertex are given) and then convert the vertex form to standard form.
The vertex form of a quadratic function has the format:
y = a Â· (x  h)^{ 2 }+ k,
where the letter h represents the xcoordinate of the vertex and the letter k represents the
ycoordinate of the vertex.
Figure 1 shows that the xcoordinate of the vertex is equal to 3 and that the ycoordinate
of the vertex is equal to 1. This means that the vertex form of this quadratic will be:
y = a Â· (x  3)^{ 2} +1.
All that remains is to find the numerical value of the constant a. To do this, you can use
the x and ycoordinates of any other point (i.e. other than the vertex) that lies on the
quadratic â€“ for example the point (0, 4) shown in Figure 3. To work out the value of a
we will plug x = 0 and y = 4 into the vertex form and then solve for a.
4 = a Â· (0  3)^{ 2} +1.
4 = a Â· 9 +1.
3 = a Â· 9.
So, the equation for the quadratic function shown in Figure 1 (expressed in vertex form)
is:
To convert this equation from vertex form to standard form, you can expand by FOILing
and then collect like terms.
(Expand the (x â€“ 3)2 by FOILing)
(Multiply through by one third)
(Combine the like terms)
So, the equation for the quadratic function shown in Figure 1 (expressed in standard
form) is:
