Post on 17-Jan-2016
Quadratic Functions Algebra III, Sec. 2.1
Objective
You will learn how to sketch and analyze graph of functions.
Important Vocabulary
Axis – the line of symmetry for a parabola
Vertex – the point where the the axis intersects the parabola
The Graph of a Quadratic Fn
A polynomial function of x with degree n is
A quadratic function is
The Graph of a Quadratic Fn (cont.)
A quadratic function is a polynomial function of _____________ degree. The graph of a quadratic function is a special “U” shaped curve called a _____________ .
second
parabola
The Graph of a Quadratic Fn (cont.)
If the leading coefficient of a quadratic function is positive, the graph of the function opens ________ and the vertex of the parabola is the ____________ y-value on the graph.
up
minimum
The Graph of a Quadratic Fn (cont.)
If the leading coefficient of a quadratic function is negative, the graph of the function opens ________ and the vertex of the parabola is the ____________ y-value on the graph.
down
maximum
The Graph of a Quadratic Fn (cont.)
The absolute value of the leading coefficient determines ____________________________________.
If |a| is small, _________________________________________________________________________________.
how wide the parabola opens
the parabola opens wider than when |a| is large
Example 1Sketch and compare the graphs of the quadratic functions.
a)
Reflects over the x-axisVertical stretch by 3/2
Example 1Sketch and compare the graphs of the quadratic functions.
b)
Vertical shrink by 5/6
Standard Form of a Quadratic Fn
The standard form of a quadratic function is ________________________________.
The axis of the associated parabola is ___________ and the vertex is ____________.
a ≠ 0
x = h (h, k)
Standard Form of a Quadratic Fn
To write a quadratic function in standard form…
Use completing the square
add and subtract the square of half the coefficient of x
Example 2Sketch the graph of f(x). Identify the vertex and axis.
Write original function.
Add & subtract (b/2)2 within parentheses.
Regroup terms.
Simplify.
Write in standard form.
Example 2Sketch the graph of f(x). Identify the vertex and axis.
a = 1 h = 5 k = 0
Vertex: (5, 0) Axis: x = 5
Standard Form of a Quadratic Fn
To find the x-intercepts of the graph…
You must solve the quadratic equation.
Example 3 Sketch the graph of f(x). Identify the vertex and x-intercepts.
Write original function.
Factor -1 out of x terms.
Add & subtract (b/2)2 within parentheses.
Regroup terms.
Simplify.
Write in standard form.
Example 3 Sketch the graph of f(x). Identify the vertex and x-intercepts.
a = -1 h = -2 k = 25
Vertex: (-2, 25) Axis: x = -2
Example 3 Sketch the graph of f(x). Identify the vertex and x-intercepts.
Set original function =0.
Factor out -1.
Factor.
Set factors =0.
Solve.
Example (on your handout)
Sketch the graph of f(x). Identify the vertex and x-intercepts.
Write original function.
Add & subtract (b/2)2 within parentheses.
Regroup terms.
Simplify.
Write in standard form.
Example (on your handout)
Sketch the graph of f(x). Identify the vertex and x-intercepts.
a = 1 h = -1 k = -9
Vertex: (-1, -9) Axis: x = -1
Example (on your handout)
Sketch the graph of f(x). Identify the vertex and x-intercepts.
Set original function =0.
Factor.
Set factors =0.
Solve.
Applications of Quadratic Fns
For a quadratic function in the form ,
the x-coordinate of the vertex is given as ___________
& the y-coordinate of the vertex is given as ________.
Example 5
The height y (in feet) of a ball thrown by a child is
given by , where x is the horizontal
distance (in feet) from where the ball is thrown.
How high is the ball when it is at its maximum
height? a = - 1/8 b = 1 c = 4
Maximum height = vertex
Example (on your handout)
Find the vertex of the parabola defined by