Exploring Transformations of Quadratics

by

Lizzy Shaughnessy

In this discussion we will focus on the following quadratic equation:

The graph of this quadratic is shown here:

(Graph 1)

• What will happen if we substitute (x-4) for all of the x's?
  • The equation for this translation is or
    • We would expect the graph would shift the graph 4 units to the right. The graph of the new equation (shown below) shows that this is correct. The new graph is the same as the original graph but just shifted to the right 4 units.

(Graph 2)

• What would we need to change to make the vertex of the original parabola into the second quadrant?
  • First, let's recall that there are several ways to write a quadratic equation. We will use standard form and vertex form.
    • Standard form is where 'a' is always the coefficient of the x-squared term, 'b' is the coefficient of the x term and 'c' is the constant term. In our original quadratic, a = 2, b = 3 and c = -4.
    • Vertex form is where 'a' is still the coefficient of the x-squared term, 'h' is x-coordinate of the vertex, and 'k' is the y-coordinate of the vertex.

   

  • From Graph 1 we can see that the vertex of the parabola is in the third quadrant. So we can keep the x-coordinate of the the vertex the same and simply make the y-coordinate of the vertex positive. This will move the graph into the second quadrant. So if we can find the vertex of the original parabola then we can vertex to easily move the parabola into the second quadrant.
  • To find the x-coordinate of the vertex we can use the formula . By substituting, we get .
  • Now we can plug the x-coordinate into the original equation to get the y-coordinate of the vertex.
    • or .
  • So the quadratic has a vertex at the point . And the vertex form of this quadratic is .
  • To move the quadratic into the second quadrant we can make the y-coordinate positive. The equation for this would be: or . The following graph shows the original quadratic and the new quadratic in the second quadrant.

(Graph 3)

• What if we wanted to flip the quadratic so that it was concave but had the same vertex? What would we need to change?
  • First, recall that the value of 'a' determines if the graph is concave up or down. If a<0, the graph is concave down, or opens downward. If a>0 then the graph is concave up, or opens up. With this in mind, we can see that it makes sense that the original function opens up.
  • Our original equation, in vertex form, is . To flip the graph so that it opens down we simply need to make a=-2. The equation that has the same vertex but is flipped down is . Here is the graph of this reflection:
    

(Graph 4)

The following graph is the original parabola and the three transformations. The equation for each parabola is seen to the left of the graph.