Download Quadratic Functions and Models

Quadratic Functions and
Models
Section 3.1
•Objectives
• Learn basic concepts about quadratic
functions and their graphs
• Apply the vertex formula
• Sketch a quadratic function
• Solve applications and model data
• Use quadratic regression to model
data.
Basics
• The quadratic is a nonlinear function.
• Its standard form is
f(x) = ax2 + bx +c
• The vertex form is
f(x) = a(x – h)2 + k
where (h,k) is the vertex
Identify the function
• Y = 5x – 1
• Y = 2x2 + 1
• Y = (3x2 + 1)2
• y = 1/(x2 – 4)
Graph of a Quadratic Function
Shape of Graph
•
•
•
•
Called a parabola
If a is +, then graph opens up
If a is -, then graph opens down
The larger |a| is, the skinnier the
graph.
• The smaller |a| is, the fatter the
graph.
Vertex
A
Axis of Symmetry
How to Find the Vertex
• Y = -3(x – 1)2 + 2
• The x-coordinate is the 1 and the ycoordinate is the 2, so (1,2)
• The negative is part of the vertex form
and not part of the vertex.
• Y = 5(x + 2)2 – 5
• Rewrite in vertex form. Y = 5(x – (2))2 – 5
• So the vertex is (-2, -5).
How to Find the Axis of Symmetry
• The axis of symmetry is a vertical
line through the vertex.
• Since it is a line, it has an equation.
• Since it is vertical, it is always in the
form of x = h (the x-value of the
vertex).
Example
• The sign of the
leading coefficient
(a) is ?
• What is the vertex?
• What is the
equation of the
axis of symmetry?
How to Find the Vertex
• When quadratic function is in
standard form, use the vertex
formula to find the vertex.
• Vertex formula
• X-coordinate found by –b/2a, where a is
the leading coefficient (x2 term) and b is
the coefficient of the x term.
• Y-coordinate found by plugging above
answer in standard form and solving for
y.
How to Find the Vertex
• Y = 3x2 – 4x + 1
• a = 3 and b = -4 and c = 1
• X-coordinate = -b/2a = -(-4)/2(3) =
4/6 =2/3
• Y-coordinate = 3(2/3)2 – 4(2/3) + 1 = 1/3
• So the vertex is (2/3, -1/3)
• Y-coordinate can also be found by
(4ac – b2)/(4a).
How to Write in Vertex Form
• Identify the leading coefficient, a.
• Find the vertex.
• Put in the vertex form.
• Example
• Y = x2 – 7x + 5
• A = 1, Vertex = (3.5, -7.25)
• So the vertex form of the function is
y = 1(x – 3.5)2 – 7.25
How to Get Formula From
Graph
• Identify vertex
(-2,-2) and plug
into vertex form.
• Pick another
obvious point, and
plug into x and y in
vertex form.
• Solve for a.
• Then put vertex
and a back in
vertex form.
How to Get Formula From
Graph
•
•
•
•
•
•
•
•
Vertex (-2,-2)
(2, 8)
8= a(2 – (-2))2 – 2
8 = a(4)2 – 2
8= 16a – 2
10 = 16a
10/16 = a = 5/8
y= 5/8(x + 2)2 – 2
ORRRRRR
You can plug the vertex and two other
points into L1 and L2 in your calculator
and do a QuadReg. This will give you a.
Suppose the vertex is (-1, 3) and the
other points on the graph are (-2, 1) and
(0,1).
Stat/edit, under L1 put -1 and -2 and 0.
Under L2 put 3 and 1 and 1.
Stat/calc, #5 QuadReg, get the a value.
Example
• Sign of the leading
coefficient?
• Vertex?
• Axis of symmetry?
• Increasing/
decreasing intervals?
#83 page 185
• A farmer has 1000 feet of fence to enclose
a rectangular area. What dimensions for
the rectangle result in the maximum area
enclosed by the fence?
• Perimeter = 2L + 2W
• 1000 = 2L + 2W
• 500 = L + W
• 500 – W = L
• Area = LW
• A = (500 – W)W
• A= 500W – W2
#83 page 185
• Area is a quadratic function with x values
of width of the enclosure and y values of
the area corresponding to that width. The
graph of the function opens downward, so
it has a maximum value…at the vertex.
• Find the vertex of the parabola, the x
value is the width and the y value is the
area.
• Plug x into 500 = L + W to get the length
• 250 feet x 250 feet