**Assignment 2: Problem 4**

**Jason P. Pickhardt**

__Problem:__

Graph the parabola:

i. Overlay a new graph replacing each x by (x - 4).

ii. Change the equation to move the vertex of the graph into the second
quadrant.

iii. Change the equation to produce a graph concave down that shares the same
vertex.

iv. Generalize . . .

__Note:__

Using a program such as Graphing Calculator it would be easy
to graph the given quadratic equation. Also, we could do some simple algebraic
manipulation to replace *x* by *(x-4)* and overlay the necessary graph.
However, when we get to the latter parts of the problem it is necessary to know
some things about transforming quadratic equations. Thus it seems necessary to
explore some quadratic functions to generalize the transformations before
completing parts i-iii. The following is a summary of the exploration I
completed in order to answer the problem.

__Exploration:__

When working with functions it is sometimes useful to begin with a parent function. In the case of our given problem, the parent function is a simple parabola described by the following:

Knowing this, we can begin to explore how to transform the parent function in order to find our desired quadratic equation. We can also begin to form some ideas for generalizing the transformations of a quadratic equation. Let’s being with increasing and decreasing the coefficient of the parent function, for instance consider the following two equations graphed using the Graphing Calculator software along with the parent function.

__Figure1__- For
further reference the parent function will be graphed in purple while
new equations will be highlighted with respect to their graphed color as shown
above.

We can see in figure 1 that increasing the coefficient of
the parent function shrinks the graph of the function while decreasing the
coefficient stretches the graph. It is also necessary to note that each of the
three graphs is figure 1 share the same overall shape as well as the same
vertex of (0,0). While this information is interesting, it is necessary to
continue our exploration. Now we can add different values of *x *to the parent function to see what
type of transformation results. The following are four different types of
equations we could write and their corresponding graphs:

__Figure 2__-
Adding values of *x* to the parent
function

It is considerably harder to evaluate what type of
transformation is happening here. However, one thing that we can conclude right
away is that there is no shrinking or stretching happening when we add a value
of *x*. Also, we can see that when we
add a positive value of *b* the graph
shifts to the left and similarly adding a negative value of *b* shifts our graph to the right. It
remains to discover what is happening to the vertex of the graph since none of
these equations shares the same vertex. After going over many more examples it
is possible to find a generalization for how the vertex is transformed. We must
consider the general equation when devising rules for
this transformation, which are as follows:

*a>0 & a<0, b>0* vertex moves by

*a>0 & a<0, b<0* vertex moves by

The last item we need to explore is adding a constant to the parent function. We can add a value of 1 or -1 to the parent function and get the following:

__Figure 3__-
Adding a constant to the parent function

It is fairly easy to see from figure 3 that adding a constant shifts the graph vertically by the value of the constant. It remains now to use this knowledge to answer the questions at hand. Thus we must first graph the original equation of. For the sake of brevity we can also complete the case where x is replaced by (x-4) in the same graph. The results are as follows:

__Figure 4__-
Graph of original equation and part (i)

We can check our original findings against what we have
graphed here. If we were to simplify the case were *x* is *(x-4)* we would come
up with:

We know instantly that the graph should shrink by a factor
of 2 given the coefficient in front of the squared term. Also, we know the
vertex will shift at the very least +16 units in the vertical direction. It now
remains to see what effect the *x* term
has on the vertex. From the formulas we came up with the middle term would
shift the vertex by (3.25, -21.125). Adding the +16 units in the y-direction we
end up with a vertex of (3.25, -5.125) Using the Graphing Calculator software
we can check and conclude that indeed this is the correct vertex. Thus, we have
shown that our findings from earlier work. Using the principles from the
exploration we can find equations to satisfy part ii and iii of the problem.
They are written and graphed as follows:

ii)

iii)

__Figure 5__-
Solutions to part ii and iii

__Conclusion:__

The most important things to take from this exploration are the transformations that take place on a quadratic equation. If we know what adding a coefficient, adding a value of x or adding a constant to the parent function of does we can graph the equation quite simply. The following is a summary of the findings of this exploration.

Given the general quadratic equation:

shrinks the graph

stretches the graph

stretches the graph and flips it concave down

shrinks the graph and flips it concave down

*a>0 & a<0, b>0* vertex moves by

*a>0 & a<0, b<0* vertex moves by

c>0 shifts the vertex up by c

c<0 shifts the vertex down by c