2.6

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Section 2.6   Quadratic functions
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A baseball is popped straight up by a batter.
The height of the ball above the ground is given
by the function y  f(t)  -16t2  64t  3,
where t is time in seconds after the ball
leaves the bat and y is in feet. Let's use our
calculator
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Let's use our calculator Y ? \Y1 -16x264x3
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Let's use our calculator Y ? \Y1 -16x264x3
Window ?
Window Value
Xmin -1
Xmax 5
Xscl 1
Ymin -10
Ymax 80
Yscl 8
Graph
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Although the path of the ball is straight up and
down, the graph of its height as a function of
time is concave down.
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The ball goes up fast at first and then more
slowly because of gravity.
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The baseball height function is an example of a
quadratic function, whose general form is
y  ax2  bx  c.  
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Finding the Zeros of a Quadratic Function
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Finding the Zeros of a Quadratic Function Back
to our baseball example, precisely when does the
ball hit the ground?
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Finding the Zeros of a Quadratic Function Back
to our baseball example, precisely when does the
ball hit the ground? Or For what value of t
does f(t)  0?
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Finding the Zeros of a Quadratic Function Back
to our baseball example, precisely when does the
ball hit the ground? Or For what value of t
does f(t)  0? Input values of t which make the
output f(t)  0 are called zeros of f.
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Let's use our calculator Y ? \Y1 -16x264x3
Window ?
Window Value
Xmin -1
Xmax 5
Xscl 1
Ymin -10
Ymax 80
Yscl 8
Graph
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Now let's use the TI to find the zeros of this
quadratic function
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2nd Trace 2 zero Left Bound ? Right
Bound? Guess?
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zero X4.0463382 Y-1E-11
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Example 1 Find the zeros of f(x)  x2 - x - 6.
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Example 1 Find the zeros of f(x)  x2 - x - 6.
Set f(x) 0 and solve by factoring x2 - x - 6
0 (x-3)(x2) 0 x 3 x -2
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Example 1 Find the zeros of f(x)  x2 - x - 6.
Let's use our calculator
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Let's use our calculator Y ? \Y1 x2-x-6
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Let's use our calculator Y ? \Y1 x2-x-6
Zoom 6 gives
Window Value
Xmin -10
Xmax 10
Xscl 1
Ymin -10
Ymax 10
Yscl 1
Graph
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Now let's use the TI to find the zeros of this
quadratic function
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2nd Trace 2 zero Left Bound ? Right
Bound? Guess?
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zero x-2 y0
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2nd Trace 2 zero Left Bound ? Right
Bound? Guess?
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zero x3 y0
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Example 3 Figure 2.29 shows a graph of
What happens if we try
to use algebra to find its zeros?
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Let's try to solve
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Conclusion?
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Conclusion? There are no real solutions, so
h has no real zeros. Look at the graph again...
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What conclusion can we draw about zeros and the
graph below?
y
x
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h has no real zeros. This corresponds to the fact
that the graph of h does not cross the x-axis.
y
x
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Let's use our calculator Y ? \Y1 (-1/2)x2-2
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Let's use our calculator Y ? \Y1 (-1/2)x2-2
Window
Window Value
Xmin -4
Xmax 4
Xscl 1
Ymin -10
Ymax 2
Yscl 1
Graph
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y
x
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2nd Trace 2 zero Left Bound ? Right
Bound? Guess?
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2nd Trace 2 zero Left Bound ? Right
Bound? Guess? ERRNO SIGN CHNG 1Quit
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Concavity and Quadratic Functions
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Concavity and Quadratic Functions Unlike a
linear function, whose graph is a straight line,
a quadratic function has a graph which is either
concave up or concave down.
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Example 4 Let f(x)  x2. Find the average rate
of change of f over the intervals of length 2
between x  -4 and x  4. What do these rates
tell you about the concavity of the graph of f ?
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Let f(x)  x2 Between x -4 x -2
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Let f(x)  x2 Between x -2 x 0
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Let f(x)  x2 Between x 0 x 2
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Let f(x)  x2 Between x 2 x 4
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Let's recap
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What do these rates tell you about the
concavity of the graph of f ?
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What do these rates tell you about the
concavity of the graph of f ? Since these rates
are increasing, we expect the graph of f to be
bending upward. Figure 2.30 confirms that the
graph is concave up.
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Let's use our calculator Y ? \Y1 x2 2nd
Mode Quit ( Vars ? Enter Enter (-2) - Vars ?
Enter Enter (-4)) / (-2 - -4) Enter
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Let's use our calculator Y ? \Y1 x2 2nd
Mode Quit ( Vars ? Enter Enter (-2) - Vars ?
Enter Enter (-4)) / (-2 - -4) Enter -6
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( Vars ? Enter Enter (-2) - Vars ? Enter Enter
(-4)) / (-2 - -4) Enter -6 ( Vars ? Enter
Enter (0) - Vars ? Enter Enter (-2)) / (0 - -2)
Enter -2
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( Vars ? Enter Enter (2) - Vars ? Enter Enter
(0)) / (2- 0) Enter 2 ( Vars ? Enter Enter (4)
- Vars ? Enter Enter (2)) / (4 - 2) Enter 6
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Example 5 A high diver jumps off a 10-meter
springboard. For h in meters and t in seconds
after the diver leaves the board, her height
above the water is in Figure 2.31 and given
by (a)  Find and interpret the domain and
range of the function and the intercepts of the
graph. (b)  Identify the concavity.
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Let's use our calculator Y ? \Y1 -4.9x28x10
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Let's use our calculator Y ? \Y1
-4.9x28x10
Window
Window Value
Xmin -2
Xmax 5
Xscl 1
Ymin -10
Ymax 15
Yscl 1
Graph
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Now let's use the TI to find the zeros of this
quadratic function. 2nd MODE
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2nd Trace 2 zero Left Bound ? Right
Bound? Guess?
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Now let's use the TI to find the zeros of this
quadratic function. Zero X -.8290322 Y 0
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2nd Trace 2 zero Left Bound ? Right
Bound? Guess?
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Now let's use the TI to find the zeros of this
quadratic function. Zero X 2.4616853 Y 0
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So, our zeros (solutions) are X -.8290322 Y
0 X 2.4616853 Y 0 Which make sense?
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Which make sense? Since t 0 X -.8290322 Y
0 X 2.4616853 Y 0
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X 2.4616853 Y 0 Domain?
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X 2.4616853 Y 0 Domain? The interval of time
the diver is in the air, namely 0  t  2.462.
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X 2.4616853 Y 0 Range?
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X 2.4616853 Y 0 Range? Given that the domain
is 0  t  2.462, what can f(t) be?
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X 2.4616853 Y 0 Range? What you see in yellow.
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X 2.4616853 Y 0 Range? What you see in
yellow. What is the maximum value of f(t)?
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2nd Trace 4 maximum Left Bound ? Right
Bound? Guess?
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X 2.4616853 Y 0 Range? What is the maximum
value of f(t)? Maximum X .81632636 Y 13.265306
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X 2.4616853 Y 0 Therefore, the range is 0
f(t) 13.265306
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What are the intercepts of the graph?
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What are the intercepts of the graph? How can we
calculate?
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What are the intercepts of the graph? How can we
calculate? We already did
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What are the intercepts of the graph? How can we
calculate? We already did t 2.4616853 f(t) 0
horiz int.
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What are the intercepts of the graph? How can we
calculate? What about?
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What are the intercepts of the graph? How can we
calculate? Substitute 0 for t in the above
equation...
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What are the intercepts of the graph? t 0, f(t)
10 vert int.
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Finally, let's identify the concavity.
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What can we say about concavity?
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What can we say about concavity? Concave
down. Let's confirm via a table...
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t (sec) h (meters) Rate of change ?h/?t
0 10  
   
0.5 12.775  
   
1.0 13.100  
   
1.5 10.975  
   
2.0 6.400  
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t (sec) h (meters) Rate of change ?h/?t
0 10  
    5.55
0.5 12.775  
    0.65
1.0 13.100  
    -4.25
1.5 10.975  
    -9.15
2.0 6.400  
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t (sec) h (meters) Rate of change ?h/?t
0 10  
    5.55
0.5 12.775  
    0.65
1.0 13.100  
    -4.25
1.5 10.975  
    -9.15
2.0 6.400 decreasing ?h/?t 
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End of Section 2.6