2.1 Quadratic Functions
You can find extra notes on this website:
http://scidiv.bcc.ctc.edu/FL/MATH105/sso0201.pdf
I. Quadratic Functions
Definition: Let “n” be a non-negative integer and let an, an-1, …, a2, a1, a0
be real numbers with an ≠ 0.
The Function
f (x) = anxn + an-1xn-1 + … + a2x2 + a1x + a0
is called a POLYNOMIAL FUNCTION of x with degree “n”
Polynomial functions are classified by degree:
f (x) = a is a constant function
f (x) = mx + b is a linear function
f (x) = ax2 + bx + c is a quadratic function where a ≠ 0 and {a, b, c } is contained in the set of Real Numbers
Parabola – a graph of a quadratic function is a special type of u-shaped curve.
Since this graph opens upward, given f (x) = ax2 + bx + c, "a" is greater than 0.
if "a" is less than 0, then the parabola is opened downward.
Vertex is the turning point and the axis of symmetry is perpendicular through the x-value of the vertex. Can be found by x = -b/(2a)
Key: using f(x) = ax2
if "a" is greater than 1, the graph is a vertical stretch of the graph y= f(x)
if 0 is less than "a" which is less than 1, the graph is a vertical shrink of the graph y = f(x)
Standard Form of a Quadratic Function:
f(x) = a (x - h)2 + k, where a ≠ 0,
the axis of symmetry is the vertical line x = h, and the vertex of the function is (h, k)
Example 1: Given the vertex (4, -1) and the point (2, 3), what is the equation of the quadratic function.
y = a (x - h)2 + k
3 = a (2 - 4)2 + (-1)
4 = a (-2)2
4 = a (4)
1 = a
y = 1 (x - 4)2 - 1
Example 2: Given the vertex is (5/2, -3/4) and a point (-2, 4), what is the equation of the quadratic function.
y = a(x - h)2 + k
4 = a(-2 - 5/2)2 + (-3/4)
4.75 = a(-4.5)2
4.75 = 20.25 a
19/81 = a
y = (19/81)(x - 5/2)2 - 3/4
II. Maximum or Minimum values:
To find the maximum or minimum, find the vertex by using
x = -b/(2a)
Example: Given C = 800 - 10x + 0.25 x2
find the minimum cost and the number of fixtures:
x = -b/(2a) = 10/((2)(.25) = 10/.5 = 20
therefore there will be 20 fixtures
C = .25 (20)2 - 10(20) + 800
C = 700
So the minimum cost is $700
You can check by graphing, and the minimum point is (20, 700)
III. Identify the vertex and the intercepts Algebraically.
f(x) = x2 + 9x + 8
Vertex is x=-b/2a = -9/2 = -4.5
f (-9) = (-4.5)2 + 9(-4.5) + 8
= 20.25 - 40.5 + 8
= -12.75
vertex is (-4.5, -12.25)
Intercepts:
Let x = 0
f (0) = 02 + 9(0) + 8 = 8
(0, 8)
Let y = 0
0 = x2 + 9x + 8
0 = (x + 8)(x + 1)
x = -8 and x = -1
so
(-8, 0) and (-1, 0)
IV. Find the equation of a quadratic
Given the two x-intercepts (-2, 0) and (10, 0), can you find the equation?
A. if the parabola opens upward, "a" is greater than 0,
y = a (x - p)(x - q)
because there can be lots of answers depending upon what the value of "a", so we will
Let a = 1
f (x) = (x - (-2))(x - 10)
= (x + 2)(x - 10)
= x2 - 10x + 2x - 20
= x2 - 8x - 20
B. if the parabola opens downward, "a" is less than 0,
y - a (x - p)(x - q)
because again, there can be lots of answers depending upon what the value of "a", so we will
Let a = -1
f(x) = -(x + 2)(x - 10)
= -(x2- 8x - 20)
= -x2 + 8x + 20
