Brenda King

Introduction and generalization:

The basic quadratic equation has the form y = ax2 + bx + c.  When a = 1, b=0, and c=0, then the equation is referred to as a parent graph.

Changing the values of a, b, and c will cause either transformations or alterations to the graph shape.

There are several types of changes which can occur to the parent graph:

1. The shape can be stretched or compressed (made wide or narrow).
2. The graph can be reflected across an axis, such as the x or y – axis.
3. The graph can be shifted left or right, a horizontal movement.
4. The graph can be shifted up or down, a vertical movement.
5. The graph can have a combination of all of these changes.

Exploration:

The following diagrams show some of the changes that can be made with various values of a, b, or c.

Graph 1

Graph 2

Graph 3

Observations:

When changes are made only to a and b, the y-intercept is shared by all the transformations (remember no change in parameter c is being made).

The vertex changes in all the graphs. See 1 and 2. When c is the only value changed, then no common point is seen, see graph 3.

Reflection:   The parabola will open upward when unknown a is positive and downward when unknown a is negative.

Stretch/compress: For larger and larger values of a, the parabola stretches (gets more narrow).  For smaller and smaller values of a, the parabola compresses, or becomes more flat.

Vertical shifts:       For larger values of c, the graph moves higher and higher in the vertical direction.  For negative values of c, the graph moves down lower and lower.

Specific Example

Using one specific example will help to demonstrate all of these changes.

Standard form: y = x2 + 2x + 3

Vertex form y = (x + 1)2 + 2

There are two ways to write a quadratic formula 1) standard form and 2) vertex form.

There is an advantage when graphing to use the vertex form of the quadratic.  In this example, both forms will be used.

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

Standard form: Y = (x-4)2 + 2(x-4) + 3 = x2 - 6x + 11

Vertex form: y = (x -3)2 + 2

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

By subtracting a value, such as 5, the graph will shift down enough to be in the fourth quadrant

Standard form: y = x2 - 6x + 11 – 5 = Y = x2 - 6x + 6

Vertex form: y = (x -3)2 + 2 - 5 =  (x -3)2 - 3

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

By replacing unknown a with a negative, the graph will be reflected and face downward.

Standard from: y = -1 x2 + 6x - 6

Vertex form: y = -1 (x -3)2 - 3

i.  Replacing with (x-4)                                                                        ii.  Moving vertically down to 4th quadrant                    iii. Reflecting with same vertex