This is a write up for problem #6 from assignment 2.

Exploration of a Parabola

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If we begin our exploration of a parabola with the second degree equation

we can examine its graph which appears below.

This is a parabola with its vertex in the third quadrant and it is concave up.

If we overlay a new graph replacing each x with ( x - 4 ), then we can see how the vertex is shifted 4 units to the right. Our original equation appears in red and our new equation appears in blue in the graph below.

It may be helpful to look at the equation of our blue graph.

If we explore the possibility of moving the graph of our parabola into the second quadrant we need to move its vertex to the left and up. Since replacing x with ( x - 4 ) in the equation of the red graph moved its vertex to the right, then replacing x with ( x + 4 ) should move its vertex to the left. Furthermore, replacing the constant term (-4) with (+2) in the equation of the red graph moves the vertex of our parabola up. If we overlay the new graph colored green we see it is in the second quadrant.

Now compare the equations of all three parabolas.

Red Parabola

Blue Parabola

GREEN Parabola

It would be logical to change our equation to produce a graph that shares the same vertex and is concave down. If we overlay the graph of the following equation in black, then we can see all four parabolas together.

Notice the green and black parabolas share the same vertex and the black parabola is concave down. Now if we expand the equation of this black parabola we can compare it to the expanded equations of our other three parabolas.

Red Parabola

Blue Parabola

Green Parabola

Black Parabola

In reaching this point in our exploration, we may want to have our students complete the square for each of the equations above in order to write the equations in the form

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