## MA 100: Precalculus

Extra Credit Project

Successfully completing this extra credit project will add up to two
point to your grade. In order to get extra credit you must show all
your work and it must be correct. (Please feel free to ask for help
on this during office hours or in Math Services. You may find the
answer in an algebra text book, but please make sure you show your own
work on the assignment.)

The standard form for the equation of a parabola is:

*f(x) = a(x-h)*^{2} + k
where *a* cannot be zero, the vertex of the parabola is
*(h,k)* and the line of mirror symmetry of the parabola is
*x = h*. The coefficient *a* determines whether or not
the graph opens up or down and how much it is stretched or "squished".

More often, the equation of a parabola is given as *g(x) =
ax*^{2} + bx + c.

Use algebraic calculation to show that every equation of the form
*ax*^{2} + bx + c can be rewritten as
*a(x-h)*^{2} + k for some easy to calculate values of
*h* and *k*. In other words, find the coordinates of
the vertex *(h,k)* of a parabola whose equation is given by
*g(x) = ax*^{2} + bx + c. To do this follow the steps
below.

- Use the FOIL method to expand the standard form equation
*f(x) = a(x-h)*^{2} + k. Then simplify the formula by
combining like terms -- for instance, *ax*^{2} + amx + nx +
d would simplify to *ax*^{2} + (am+n)x + d.
- You have now written the formula for
*f(x)* in
polynomial form, so you can compare it to the formula for
*g(x)*. Set *b* equal to the coefficient of *x*
in your formula for *f(x)* and set *c* equal to the
constant term.
- These two equations describe
*b* and *c* in terms
of *h* and *k*. You wish to describe *h* and
*k* in terms of *b* and *c*. Use algebra to
solve for *h*; your answer should look similar to part of the
quadratic formula. (Why? Can you also use algebra to solve for
*k*?)
- You know that in the original equation of
*f(x)*,
*h* was the *x*-coordinate of the vertex of the graph of
*f(x)* and that the line of symmetry of the parabola was *x
= h*. Your formula for *h* in terms of the coefficients of
*g(x)* gives the *x*-coordinate of the vertex and the
line of symmetry of the graph of *g(x)*.
- Use the fact that the vertex of the parabola described by
*g(x)* will have coordinates *(h, g(h))* to describe the
coordinates of the vertex of the graph of *g(x)* in terms of
*a*, *b* and *c*.
- Conclude that
*g(x) = f(x)* when *h* and *k*
have the values you've calculated.

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