## Sunday, July 21, 2013

### ALGEBRA II: QUADRATIC EQUATIONS AND THE PARABOLA

EVERYTHING YOU NEED TO KNOW ABOUT
IN THE ALGEBRA II CURRICULUM

Quadratic equations can be expressed in several forms, each emphasizing a different aspect of graphs.  In Algebra I we learned the General Form and practiced factoring, substituting various values of X or Y, the Quadratic Formula, and (maybe) Completing the Square.  In advanced Algebra courses, we manipulate the General Form to openly express some of the familiar coordinates (like the vertex) and to examine in greater depth some of the properties of the Parabola.

While what follows are simple explanations of a Parabola that opens up, a few transformations will describe graphs which open down, right, or left.  If more detail on transformations is needed for your class, use the comments section to request examples.

GENERAL EQUATION FORM            AX^2 + BX + C = Y

VERTEX FORM                                  a (X-h)^2 -k = Y

STANDARD FORM                             X^2 = 4PY
(X-h)^2 = 4P (Y-k)

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### GENERAL FORM

AX^2 + BX + C = Y

This is the form that is used in systems of equations and matrices.

Coefficients from the General Form are also used in the quadratic formula to find X intercepts.

These X values give the intercepts of the X axis -- the roots.

The X value of the Vertex is the midpoint between the two roots.  Substitute to find the Y coordinate of the Vertex.

The axis of symmetry is .... X = the X value of the vertex.

The Y intercept is C.

#### Graphing from the General Form

Factoring or solving for the root is generally the first step in graphing from the General Form.  And, of course, the y-intercept is expressly given as the constant.

Finding the vertex takes a little more calculation.

X^2 - 10X + 21 = Y

X = 7, 3

Vertex X = 5
Vertex Y = -4

Y intercept = 21

### VERTEX FORM

a (x-h)^2 - k = y

In this form, (h,k) is the vertex of the parabola; hence the name “vertex form.”

To move from the General Form to the Vertex Form, you will need to complete the square.  (See the blog, Quick Steps to Completing the Square, 7-22-13)

To find the roots (solutions), solve for X by isolating the (x-h)^2, taking the square root of both sides, and isolating  X.

The y intercept is calculated by substituting 0 (zero) for X.

#### Graphing from the Vertex Form

As the name implies, the vertex is the first step in graphing from this form.  Calculations are needed to find the axis intercepts.

a (x-h)^2 -k = y

Vertex = (5, -4)

X intercepts = (7,0) and (3,0)

Y intercept = 21

Did you notice that the same 4 points are used to graph from both the General and Vertex Forms?  If a fifth point is required for class, use the symmetry principle to find the point directly across from the y-intercept and the same distance from the axis of symmetry but on the other side.

### STANDARD FORM

X^2 = 4PY

(x-h)^2 = 4P (y-k)

This form emphasizes the Focus and Directrix of the graph.  (h,k) is the vertex and P is the distance from the vertex to the Focus point along the axis of symmetry (X = Vertex X) and the distance from the Vertex to the Directrix.  The Directrix is the horizontal line, Y = Vertex Y - P.

To solve for the roots, substitute 0 (zero) for Y.

To solve for the Y intercept, substitute 0 (zero) for X.

#### Graphing from the Standard Form

Details represented in this form are especially useful in physics.  The focus is the point to which any ray striking the ‘”cup” of the parabola is reflected.

(X - 5)^2 = 1(Y + 4)
4P = 1
P = 1/4

Vertex = (5, -4)
Focus = (5, -15/4)
Directrix, Y = -17/4
X intercepts = (3,0), (7,0)
Y intercept = (0,21)

Do you notice anything “special” about the Standard Form compared with the Vertex Form?  They are actually the same, but the Standard Form specifically mentions the 4P value.  A little algebra and you’ll see that  4P = 1/ a.  All we’ve really done is isolate the squared binomial by moving everything else to the other side of the equation.

Any of these equations can describe a Parabola opening down by changing P to the Arithmetic Inverse or a Parabola opening left or right by exchanging X with Y and vice versa.

### Relation of the Focus, Directrix, and Latus Rectum*

Any point on the Parabola is equidistant from the Focus and the Directrix; that’s the definition of the Parabola, and the distance is explicitely expressed in the Standard Form as P.  The Latus Rectum is the length of a line perpendicular to the Axis of Symmetry, through the focus, and intersecting the curve of the Parabola.  It is also directly expressed in the Standard Form as 4P.

*  Please excuse a little silliness in the middle of all this serious math.  I work with Middle Schoolers quite a bit and, although many schools exclude it from the curriculum, mention of the “Latus RECTUM” always brings a giggle, something that is too often missing in the math classroom.