1. Chapter 3 Quadratic Functions

2. 3.1 Investigating Quadratic Functions in Vertex Form
A quadratic function is a function (f) whose value f(x) at x is given by a polynomial of a degree 2.
( ex. f(x) = x2 )
The graph of a quadratic function is called the parabola (a symmetrical curve). When the graph opens upwards, the vertex is the lowest point of the graph and has a minimum value. When the graph opens downwards, the vertex is the highest point on the graph, and has a maximum value.
The parabola also has an axis of symmetry, which is a line through the vertex that divides the graph into two congruent halves. Also the x-coordinate of the vertex corresponds to the equation of the axis of symmetry.
Quadratic functions are written in vertex form that tells you the location of the vertex: (p, q), the shape of the parabola and the direction of opening.
The formula is: f(x) = a(x - p)2 + q
a determines the orientation and shape of the parabola
the graph opens upwards if a > 0 and downward if a < 0
if -1 < a < 1, the parabola is wider compared to the graph of f(x) = x2
if a > 1 or a < -1, the parabola is narrower compared to the graph of f(x) = x2

3. Example:
Determine the following characteristics for the function: the vertex, the domain and range, the direction of opening, and the axis of symmetry. y = 2(x + 1)2 - 3
Since p = -1 and q = -3, the vertex is then located at (-1, -3).
Since a > 0, the graph opens upwards. Also, a > 1, the parabola is narrow.
Since q = -3, the range is { y | y ≥ -3, y ∈ R }
The domain is { x | x ∈ R }.
Since p = -1, the axis of symmetry is x = -1.

4. 3.2 Investigating Quadratic Functions in Standard Form
The standard form of a quadratic function is f(x) = ax2 + bx + c or y = ax2 + bx + c , where a, b, and c are real numbers and a ≠ 0.
a determines the shape and the direction of opening
b influences the horizontal position of the graph
c determines the y-intercept of the graph
In this section, a way to find the vertex form when given the standard form is:
b = -2ap or p = -b and c = ap2 + q or q = c - ap2 2a
Recall that to determine the x-coordinate of the vertex, you can use the equation x = p. So the coordinate of the vertex is x = -b

5. 3.3 Completing the Square Method of Completing the Square:
Another way of finding the vertex form when given the standard form is by completing the square. Completing the square is an algebraic process used to write a quadratic polynomial in the form
a(x - p)2 + q.
Method of Completing the Square:
factor a (the coefficient of x2) out of the right side of the equation.
add and subtract the square of half the coefficient of the x-term to both sides of the equation
group the square trinomial (factor the right side of the equation -there will always be two identical factors-)
solve the equation for y, so that the equation is in vertex form

6. Example 1: Example 2: Complete the square of y = x2 - 8x + 5
y = (x2 - 8x) Group the first two terms
y = (x2 - 8x + 16) Add & subtract the square of half the coefficient of the x term
y = (x- 4) Rewrite as the square of a binomial
y = (x- 4) Simplify
Example 2:
Complete the square of y = 3x2 -12x -9
y = 3(x2 - 4x) -9
y = 3(x2 - 4x) -9 Only factor out the first two terms
y = 3(x2 - 4x ) -9
y = 3(x2 - 4x + 4) Multiply 4 by 3
y = 3(x2 - 4x + 4) -21
y = 3(x - 2) Rewrite as the square of the binomial