Quadratics are written in three basic forms. They each have their own uses. The names for the three forms are unimportant and may depend on the textbook, teacher or country. **The IB does not give names for the different forms.** The names I choose I like and make sense to me…

#### Standard/General Form

This is the form that most students are comfortable with, but is one of the least useful forms.

(1)By least useful I mean that it is challenging to make an accurate sketch of the function from this form. The *c* value represents the *y-intercept* and *a* represents a vertical dilation neither of which allow for easy graphing. Plus *b* does "strange" things.

This form is handy because the parameters (*a,b,c*) correspond to the parameters in the quadratic formula:

It is also very common to make use of the formula for the axis of symmetry in IB problems which also makes use of the parameters in the standard form:

(3)#### Factored Form

This form appears like:

(4)Where *p* and *r* are the zeros of the function. Graphically the zeros are where the function crosses the x-axis. The sign of *a* tells you which direction the parabola opens. Knowing the points that the graph crosses the *x-axis* and which way the parabola opens allows a quick and moderately accurate sketch of the function.

#### Vertex Form

This form appears like:

(5)In this form *h* represents a horizontal shift and *k* represents a vertical shift. If both *h* and *k* are zero then the vertex of the parabola would be at $(0,0)$. There for the coordinates of $(h,k)$ give us the vertex of the parabola! The sign of *a* tells you which direction the parabola opens

#### A Comment on All Forms

In all the forms *a* plays the same role. In fact if you move from one form to another *a* will always have the same value, which is a very handy thing to remember.

#### Converting Between Forms

### Getting to Vertex Form

Getting into Vertex Form often seems to be the trickiest bit for students. Below are explanations how to do it with and without your GDC.

#### Example - Without GDC

Write the function $2x^2+5x+3$ in the form $a(x-h)^2+k$.

For the example without GDC for getting to vertex form, shouldn't the equation be {2(x+1.2)^2}-0.125 if the x value of the vertex is -1.2?

ReplyOptionsYes, if x is equivalent to h and x = -1.2

then wouldn't it be y = 2(x+1.2)2 - 0.125

as the two negatives make it a positive?

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