Quadratic Forms

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.

\begin{equation} y=ax^2+bx+c \end{equation}

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:

\begin{align} x=\frac{-b \pm \sqrt{b^2-4ac}}{2a} \end{align}

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:

\begin{align} x=\frac {-b}{2a} \end{align}

Factored Form

This form appears like:

\begin{equation} y=a(x-p)(x-r) \end{equation}

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:

\begin{equation} y=a(x-h)^2+k \end{equation}

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 $a(x-h)^2+k$.

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