Vertex Form of the Quadratic

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Vertex and Intercept Form of
Quadratic Function
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.
Vertex Form of the Quadratic
Determine the vertex of the following
functions:
•f(x) = 2(x – 1)2 + 1
•g(x) = -(x + 3)2 + 5
•h(x) = 3(x – 2)2 – 7
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

Vertex Form of the Quadratic
Determine the vertex of the following
equations:
f(x) = 2(x – 1)2 + 1 V = (1, 1)
g(x) = -(x + 3)2 + 5 V = (-3, 5)
h(x) = 3(x – 2)2 – 7 V = (2, -7)
Compare
equations
and
theofvertices.
The x partthe
is the
opposite
sign
the
Do
you notice
pattern?
number
insideathe
brackets and the y part is
the same as the number added or
subtracted at the end.
Vertex Form of the Quadratic
 The
vertex form of the quadratic
equation is of the form:
 y = a(x – h)2 + k, where:
 The
vertex is located at (h, k)
 The axis of symmetry is x = h
 The “a” is the same as in the standard form
 The “a” is the stretch of the function
 The vertex is shifted right by h
 The vertex is shifted up by k
Vertex Form of the Quadratic
From y = x2
Stretch factor
Vertex Shift VERTICAL
amount
y = a(x –
2
h)
+k
Vertex Shift HORIZONTAL amount
6
5
fx = x+32
a>0
Graph Opens UP
Vertex is a MINIMUM
4
Parent Curve:
3
hx = x2
2
gx = -x-32+5
1
-6
-4
-2
a<0
Graph Opens DOWN
Vertex is a MAXIMUM
2
-1
4
6
In Class:
 Do
page 63 of Note Taking Guide
 Do first 6 problems of Henley Task Day
2 – be sure to graph the y = x2 for each
graph.
In Class
 Do
page 64 of the Note Taking Guide
 Do Day 2 of the Henley Task, # 4a – 4e
all
Intercept Form of the
Quadratic Function
How can we determine the vertex of the
following equations without putting them in
standard form?
V = (2, -1)
•f(x) = (x – 3)(x – 1)
•g(x)
= 2(x + 1)(x + 4) V = (-2.5, -4.5)
•h(x) = -3(x – 2)(x + 3) V = (-0.5, 18.75)
•Determine the x-intercepts (zero prod rule)
•Find the axis of symmetry (average)
•Find “y” value of the vertex (sub into f(x))
Homework
 Page
65, # 1, 2, and 19 – 22 all
Convert from Standard to
Vertex Form
Standard: MM2A3a Students will
Convert between standard and vertex
form.
Convert from Standard to
Vertex Forms
 We
converted from Vertex form to
Standard form of the quadratic function
above in slide 3 by expanding the
(a – h)2 term and combining like terms
 How can we convert from Standard
form to Vertex form?
Convert from Standard to
Vertex Forms
 Look
at the standard form:
y = ax2 + bx + c, where a ≠ 0
 And look at the Vertex form:
y = a(x – h)2 + k
 “h” is the axis of symmetry, which is the
“x” part of the coordinates of the vertex
 “k” is the “y” part of the vertex
Convert from Standard to
Vertex Forms
 How
did we find the axis of symmetry?
 b This is the “h” of the vertex

x
2a form
 How did we then find the “y” part of the
vertex?
 Substitute the x into the original
equation and solve for y.
 This is the “k” of the vertex form
 The “a” is the same for both forms
Convert from Standard to
Vertex Forms
 Convert
the following functions to vertex
form:
 f(x) = x2 + 10x – 20
 y = (x + 5)2 - 45
 g(x) = -3x2 – 3x + 10
 y = -3(x + 0.5)2 + 10.75
 h(x) = 0.5x2 – 4x – 3
 y = 0.5(x – 4)2 - 11

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