04-2b+-+Standard+Form+of+a+Quadratic+Equation

Standard Form

We can derive the standard form of a quadratic function from the vertex form which we used when discussing the axis of symmetry. Recall that the standard form of a quadratic function is  where.

Let's look at this derivation.

Vertex Form

 expand the binomial  distribute "a"  Simplify and group terms



So we see that "a" in standard form is the same as in vertex form -- it tells us whether a vertical stretch or compression has been applied.

"-2ah" is the coefficient of the linear term, x,and is equivalent to "b" in standard form.

Notice that the sign of the x term in the equation is positive -- if you see a negative sign in front of the x-term, it belongs to the coefficient. If we solve -2ah = b for h, we get which is the equation for the axis of symmetry.

The constant term, c, is equal to ah 2 + k, which is the same as what we see in the vertex form of f when x = 0 :

f(0) = a(0 - h) 2 + k

From this we see that c is the y-intercept.

We can generalize the propeties of the standard form and the vertex form of the quadratic function to help us graph quadratic functions

.

Here's a little hint -- when a is positive, the parabola is happy ( U) When a is negative, the parabola is sad

Let's try an example:

Consider the function f(x) = 2x2 - 4x + 5

Does the graph open upward or downward? Since a is positive, the graph opens upward. What is the axis of symmetry? Using the equation and substituting -4 for b and 2 for a, we find that the equation of the axis of symmetry is x = 1.

Where is the vertex? The vertex always lies on the axis of symmetry, so the x-coordinate is 1. The y coordinate is the value of the function at x=1, so substitute into the equation: f(1) = 2(1) 2 - 4(1) + 5 = 3

The vertex is at (1, 3).


 * What is the y-intercept? Since c = 5, the y-intercept is 5.
 * What does the graph look like?
 * <span style="display: block; font-family: Verdana,Geneva,sans-serif;">First, sketch the axis of symmetry, then plot the vertex and the intercept.



<span style="font-family: Verdana,Geneva,sans-serif;">

<span style="display: block; font-family: Verdana,Geneva,sans-serif;">Finally, sketch the curve. We know the general shape and we have 3 points that the curve must pass through. You can calculate the coordinates of additional points to help make the curve smoother, or you can enter the equation into a grapher.



<span style="display: block; font-family: Verdana,Geneva,sans-serif;">Since this graph was drawn by a program, it is much more accurate than one drawn by hand, so you can feel confident in reading the coordinates of points off this graph. <span style="display: block; font-family: Verdana,Geneva,sans-serif;"> <span style="display: block; font-family: Verdana,Geneva,sans-serif;">Proceed to Domain, Range, Maxima and Minima