Transcript Slide 1

•Henley
Task teaches horizontal
transformations
•Protein Bar Toss Part 1 teaches factoring if
a≠1
•
•
Section 3.4 for a = 1
Section 3.5 for a ≠ 1
•Protein
Bar Toss Part 2 teaches changing
from standard and vertex forms
Math II, Sections 3.1 – 2.4
Standard: MM2A3c Students will
Investigate and explain characteristics of
quadratic functions, including domain,
range, vertex, axis of symmetry,
zeros, intercepts, extrema,
intervals of increase and
decrease, and rates of
change.
Quadratic Function
A
quadratic function is a function that
can be written in the standard form:
y = ax2 + bx + c, where a ≠ 0
 The graph of a quadratic equation is a
parabola. The lowest or highest point
on a parabola is the vertex. The axis of
symmetry divides the parabola into
mirror images and passes through the
vertex.
A
quadratic function is a function
 What
is a function?
 The
graph of a quadratic equation is a
parabola
 What
is a parabola, what does it look like?
 The
lowest or highest point on a
parabola is the vertex
 What
does it mean when we say the lowest
or highest point?
 The
axis of symmetry divides the
parabola into mirror images
 What
are mirror images
Axis of Symmetry of Quadratic
 The
quadratic function is a symmetrical
function around a vertical axis of
symmetry. That means, if we draw a
vertical line through the function, the
distance from the axis of symmetry to
the function in both directions is the
same.
Axis of Symmetry of Quadratic
 Graph
(using an “H” table), calculate the
zeros and compare axis of symmetry of
the following functions:
f(x) = x2 + 6x + 5
g(x) = x2 + 6x + 9
h(x) = x2 + 6x - 7 (later)
Show some Geosketch examples
 Explain your findings and make a
statement about what c does in the
equation of ax2 + bx + c = 0
Calculating the Axis of
Symmetry of Quadratics
 The
c in the standard form of the
quadratic equation ax2 + bx + c = 0,
simply moves the graph vertically. It
does not change the axis of symmetry.
 Since c can be changed without
changing the axis of symmetry, let us
choose c to equal zero and find the
zeros of the resulting equation and the
axis of symmetry.
Axis of Symmetry of Quadratic
now have: ax2 + bx = 0
 Factoring out GCF gives: x(ax + b) = 0
 Solving gives x = 0 or x = -b/a
 We also know the axis of symmetry is
the vertical line in the center of the
zeros, so the axis of symmetry is at the
mean (average) of the two zeros.
 The axis of symmetry is located at:
 We
x = (–b/a + 0)/2 = -b/(2a)
Axis of Symmetry of Quadratic
 Determine
the equation for the axis of
symmetry for our equations:
f(x) = x2 + 6x + 5
g(x) = x2 + 6x + 9
h(x) = x2 + 6x - 7
 The equation is x = -3
 Draw a vertical line through x = -3
 Calculate the distance on the x-axis
from the axis of symmetry to each zero.
 Explain what you notice.
Axis of Symmetry of Quadratic
 Can
we use the same equation to
determine the axis of symmetry for
functions that do not cross the x-axis?
Graph and determine the equation for
the axis of symmetry for:
h(x) = x2 + 6x + 12
 Use the line y = 7 to determine the
distance from the axis to the function.
 Explain your results.
Axis of Symmetry of Quadratic
 The
axis of symmetry can still be
determined by x = (0 –b/a)/2 = -b/(2a)
even for functions that do not cross the
x-axis.
New Graph
 Graph
and find axis of symmetry:
i(x) = -x2 + 6x – 8 (a new function)
Location of the Vertex
 Look
at our graphs & equations again:
f(x) = x2 + 6x + 5
g(x) = x2 + 6x + 9
i(x) = -x2 + 6x – 8 (the new function)
 Explain how can we find the coordinates
of the vertices?
 Determine the general equation for the
coordinates of the vertices.

Vertex is at (-b/2a, f(-b/2a))
Vertex & Axis of Symmetry Summary
Put equation in standard form f(x) = ax2 + bx + c
 Determine the value “a” and “b”
 Determine if the graph opens up (a > 0) or down
(a < 0)
b
 Find the axis of symmetry:

x
2a
Find the vertex by substituting the “x” into the
function and solving for “y”
 Determine two more points on the same side of
the axis of symmetry
 Graph the axis of symmetry, vertex, & points

Practice: Graphing, Vertex,
Axis of Symmetry
58, # 1 – 4 all,
 Page 59, # 23 – 34 all
 (do some in class together)
 Page
End Conditions, Max/Min
 Look
at our graphs & equations again:
f(x) = x2 + 6x + 5
g(x) = x2 + 6x + 9
i(x) = -x2 + 6x – 8 (a new function)
 What are their end conditions?
 Do they have a maximum or minimum?
 Explain how we can tell the end
conditions and if a function has a
maximum or minimum from looking at
the equation.
Domain of a Quadratic
Function
 Look
at our graphs & equations again:
f(x) = x2 + 6x + 5
g(x) = x2 + 6x + 9
i(x) = -x2 + 6x – 8
 What is the domain of each equation?
 What general rule can we make about
the domain of a quadratic function
 The domain of a quadratic equation is
all real numbers
Range of a Quadratic
Function
 Look
at our graphs & equations again:
f(x) = x2 + 6x + 5
g(x) = x2 + 6x + 9
i(x) = -x2 + 6x – 8
 What is the range (values of y) of each
equation?
 Does the range differ whether a is
positive or negative?
 What general rule can we make about
the range of a quadratic function?
Range of a Quadratic
Function
 Look
at our graphs & equations again:
f(x) = x2 + 6x + 5
g(x) = x2 + 6x + 9
i(x) = -x2 + 6x – 8
 If a is positive, the range is:
y = {y | y  f(-b/2a)}
 If a is negative, the range is:
y = {y | y  f(-b/2a)}
Practice: Graphing, Vertex,
Axis of Symmetry, Min/Max,
Open Up/Down, Domain &
Range
58 & 59, # 5 – 22 all
 Page 59, # 35 – 41 all
 Page
Intervals of Increasing and
Decreasing
 Look
at our graphs & equations again:
f(x) = x2 + 6x + 5
g(x) = x2 + 6x + 9
i(x) = -x2 + 6x – 8
 Over what intervals are the functions
increasing?
 Over what intervals are the functions
decreasing?
 Explain how the sign of a affects the
rules of increasing and decreasing.
Rates of Change (3.3)
 Look
at our graphs & equations again:
f(x) = x2 + 6x + 5
g(x) = x2 + 6x + 9
i(x) = -x2 + 6x – 8
 Slope of a linear function is defined as
rise/run = (y2 – y1)/(x2 – x1)
 These functions are not linear. How can
we talk about the slope of these functions?
 Explain how the slope of the functions
change as we move across the domain.
Practice Rate of Change
 Pg
72, # ?? - ??
Summary
 For
all quadratics:
 Axis
of symmetry is at x = -b/2a
 Vertex is at (-b/2a, f(-b/2a)
 The vertex is the extreme
 Domain (x) is all real numbers
 The zeros, intercepts, solutions, are the
determined by moving everything to one
side of the equation (equal zero), factoring,
and solving via the zero product rule.
Summary
 If
a>0
 Parabola
opens up
 Vertex is at the minimum
 Rise to the left and right
 Range (y) is all real numbers  -b/2a
 Rate of change is zero at the vertex, and
becomes more negative as x decreases,
and more positive as x increases
 Intervals of increasing x  vertex
 Intervals of decreasing x  vertex
Summary
 If
a < 0 (the opposite of a > 0)
 Parabola
opens down
 Vertex is at the maximum
 Fall to the left and right
 Range (y) is all real numbers  -b/2a
 Rate of change is zero at the vertex, and
becomes more negative as x increases,
and more positive as x decreases
 Intervals of increasing x  vertex
 Intervals of decreasing x  vertex