Download Chapter 3 Notes - Mr Rodgers` Math

Chapter 3 Quadratic
Functions
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
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.
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
2a
and
c = ap2 + q
or
q = c - ap2
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
2a
3.3 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
Example 1:
Complete the square of y = x2 - 8x + 5
y = x2 - 8x + 5
y = (x2 - 8x)
+5
2
y = (x - 8x + 16) + 5 - 16
y = (x- 4)2 + 5 - 16
y = (x- 4)2 - 11
Group the first two terms
Add & subtract the square of half the coefficient of the x term
Rewrite as the square of a binomial
Simplify
Example 2:
Complete the square of y = 3x2 -12x -9
y = 3(x2 - 4x) -9
y = 3(x2 - 4x) -9
y = 3(x2 - 4x
) -9
2
y = 3(x - 4x + 4) -9 - 12
y = 3(x2 - 4x + 4) -21
y = 3(x - 2)2 -21
Only factor out the first two terms
Multiply 4 by 3
Rewrite as the square of the binomial