Download 8.3 Exploration???

8.3 Graphing 𝒇(𝒙) = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄
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Exploration:
5.5
Recall: One way to find the number that is exactly halfway between two given numbers is to add up the
3
1
two given numbers and then multiply by
. Use this method to find the number that is halfway
2
between the numbers 4 and 20. Show your work.
Investigation: The graph of the equation
y  x 2  6x
is given below.
a) Using the graph, find the x-intercepts and the x-coordinate of the vertex of the parabola.
x-intercepts:
x
and
x
x-coordinate of the vertex:
x
b) You should see a graphical and a mathematical relationship between the x-coordinate of the
vertex and the x-intercepts. Describe this relationship.
c) Do you think this relationship will be true for all parabolas? Why?
d) Find the x-intercepts of the graph of the same equation
algebraically by factoring and solving for x when
y  x 2  6 x , but this time do it
y  0.
e) How can we find the x-coordinate of the vertex algebraically? (Explain)
8.3 Graphing 𝒇(𝒙) = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄
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Extension:
a) Find the x-intercepts of the equation
x-intercepts:
x
and
5.5
y  ax 2  bx
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algebraically.
x
b) Use these x-intercepts to find the x-coordinate of the vertex for
y  ax 2  bx .
(Note: Your answer will be a general formula involving a and b)
x-coordinate of the vertex:
x
c) Note that the quadratic function
Pick out the values of a, b, and c
y  3x 2  12 x  5 is in the form y  ax 2  bx  c .
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for y  3x  12 x  5 and write them below.
b
a
c
Now use your formula from part (B) to find the x-coordinate of the vertex for the parabola
graph of the equation
y  3x 2  12 x  5 .
x-coordinate of the vertex:
x
d) Find the y-intercept of the graph of
y  3x 2  12 x  5 algebraically. (Find the value of
x  0 ).
e) Will the y-intercept for a quadratic
y  ax 2  bx  c
always be the value of c ? Explain.
y when
8.3 Graphing 𝒇(𝒙) = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄
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If a is POSITIVE ( a > 0 )
The parabola will open UPWARD
.
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The vertex gives the MINIMUM value.
There is NO MAXIMUM value.
The graph of a quadratic function
If a is NEGATIVE ( a < 0 )
5.5
The parabola will open DOWNWARD
.
The vertex gives the MAXIMUM value.
There is NO MINIMUM value.
y  ax 2  bx  c
is a parabola:
The y-intercept will be the value of c .
The x-coordinate of the vertex will be
The axis of symmetry will also be
x
x
b
2a
b
2a
(Since the vertex lies on this axis of symmetry)