Unit 8 -- Quadratic Functions
Lesson 3
CCSS A.CED.2
CCSS F.IF.4
CCSS F.IF.7a
I can...create equations in two or more variables to represent relationships
between quantities.
I can... interpret key features of a graph.
I can...graph quadratic functions and identify the intercepts, maxima, and
minima .
Warm up
1) Give two numbers whose product is 55 and whose sum is 16.
answer: 11 and 5
2)
answer: D
Essential Question:
How can you obtain the graph of g(x) = a(x-h)2 + k from the graph of:
Engage:
f(x) = x2 ?
g(x) = 2(x-3)2 + 1
stretch the graph by a
graph f(x) =x2
scale factor of 2 to get
the graph of f(x)
translate the graph of
f(x) = 2x2 right 3 units
and up 1 unit to obtain
the graph of
f(x) = 2(x-3)2 + 1
Reflect:
1a) The vertex of the graph of f(x) = x2 is
_______
The vertex of the graph of g(x)=2(x-3)2 + 1 is ______.
2a) What can you tell about the graph of g(x) = -2(x-1)2 + 3
without graphing?
Let's Graph
g(x) = -3(x+1)2-2
Identify and plot the vertex:
Vertex:
parent function in red
Identify and plot other points based
on the parent function f(x) =x2.
Reflect:
2a) List the transformations of the graph of the parent function
f(x) = x2, in the order that you would perform them, to obtain the
graph of g(x) = -3(x+1)2 - 2.
2b) if you change the -3 in the equation to 4, how would the graph
change?
Writing a Quadratic Function from a Graph
A house painter standing on a ladder drops a paintbrush, which falls
to the ground. The paintbrush's height above the ground (in feet) is
given by a function of the form f(t) = a(t-h)2 + k where t is the time
(in seconds) since the paintbrush was dropped.
Because f(t) is a quadratic function,its graph is a ____________.
Only the portion of the parabola that lies in Quadrant I are valid for
the domain and range -- why?
Graph of the situation:
Vertex is: (
,
(0,30)
)
Substitute the values of h and k into the
general equation for f(t) to get
f(t) = a(t-
(1,14)
)2 +
From the graph f(1) = ___
Substitute 1 for t and ____ for f(t) and solve for a.
____ = a(1 - 0)2 + 30
The equation is: _________
Reflect:
Using the graph (or the equation), estimate how much time elapses
until the paintbrush hits the ground t = ____________
Why doesn't f(2) make sense in this situation?
Extra example:
A driver jumps off a cliff into the sea below to
search for shells. The diver's height above the sea (in
(0,40)
feet) is given by a function of the form
f(t) = a(t-h)2 + k,
where t is the time(in seconds) since the diver
jumped. Use the graph to find an equation for f(t).
Vertex: ____________
(1,24)
f(t) = a(t-h)2 + k
plug in (1,24) for t and f(t), plug in vertex for h and k, then
solve for a and write the equaition.
Exit Ticket:
Given g(x) = -2(x-1)2 + 3
Without graphing...
name: _______________
period: ______
1) State the vertex of the function.
2) State whether the parabola opens up or down.
3) Is this graph (skinnier-vertical stretch or wider-vertical shrink or the
same width - no stretch or shrink) than the parent function
f(x) = x2.
4) Compared to the parent function f(x) = x2, how is this function
shifted (left, right, up, down)?
Practice or Homework:
Graph each quadratic function.
1) f(x) = 2(x-2)2 + 3
3) f(x) = 1/2(x-2)2
2) f(x) = -1(x-1)2 +2
4) f(x) = -(1/3)x2 - 3
5) A roofer working on a roof accidentally drops a hammer, which falls to the
ground. The hammer's height above the ground(in feet) is given by a function of
the form f(t) = a(t-h)2 + k where
t is the time (in seconds) since the hammer was dropped.
Because f(t) is a quadratic function, its graph is a parabola. Only the portion of the parabola that lies in Quadrant
I and on the axes is shown because only nonnegative values of t and f(t) make sense in this situation. The vertex
of the parabola lies on the vertical axis (the y-axis).
a) use the graph to find an equation for f(t).
(0,45)
(1,29)
b) estimate how long it will take the hammer
to reach the ground?
0
2
Practice or Homework:
Graph each quadratic function.
1) f(x) = 2(x-2)2 + 3
2) f(x) = -1(x-1)2 +2
3) f(x) = 1/2(x-2)2
4) f(x) = -(1/3)x2 - 3
5) A roofer working on a roof accidentally drops a hammer, which falls to the
ground. The hammer's height above the ground(in feet) is given by a function of
the form f(t) = a(t-h)2 + k where
t is the time (in seconds) since the hammer was dropped.
Because f(t) is a quadratic function, its graph is a parabola. Only the portion of the parabola that lies in Quadrant
I and on the axes is shown because only nonnegative values of t and f(t) make sense in this situation. The vertex
of the parabola lies on the vertical axis (the y-axis).
a) use the graph to find an equation for f(t).
(0,45)
h(x) = -16t2 + 45
(1,29)
b) estimate how long it will take the hammer
to reach the ground?
0 = -16t2 + 45
-45 = -16t2
note to teacher:
2.8125 = t2
t is approximately 1.7
Student may either find this algebraically or by looking at their graph.
0
2