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3.5 Maximum and Minimum Values
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A quadratic function is a function which can be written in the form
f (x ) = ax2 + bx + c
( a ≠ 0 ). Its graph is a parabola.
Every quadratic function f (x ) = ax2 + bx + c
can be written in
standard form: f (x ) = a( x − h)2 + k . The vertex is (h, k ) .
f ( x ) = x2
f ( x ) = −4 x2
For f ( x) = ax 2 + bx + c :
• The graph opens down if a < 0 .
For Shortcut:
 b  b 
f ( x ) = ax2 + bx + c , the vertex is  −
, f −
  . So the axis of
 2a  2 a  
b
symmetry is x = −
.
2a
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3.5 Maximum and Minimum Values
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For a quadratic function f ( x ) = ax2 + bx + c , or f ( x ) = a( x − h)2 + k :
• The graph opens up if a>0. That indicates a minimum.
• The graph opens down if a < 0. That indicates a
maximum.
The vertex is the lowest or highest point of a quadratic
function, so the minimum or maximum is given by the y-value of
the vertex. By completing the square on the quadratic function
the equation can be put into standard from.
Take the equation f (x ) = ax 2 + bx + c and put in standard form
which will give us the information needed to answer several
questions. To do this by completing the square to get standard
form f (x ) = a( x − h)2 + k . The point (h, k) is the vertex. Take note
that you change the sign of the x coordinate h.
Example 1:
Write the equation in standard form, state the coordinates of
the vertex, sketch the graph and state the maximum or
minimum value of the function.
f (x) = x2 + 4 x + 1
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3.5 Maximum and Minimum Values
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Example 2:
Write the equation in standard form, state the coordinates of
the vertex, sketch the graph and state the maximum or
minimum value of the function.
f ( x ) = −3 x 2 + 6 x + 2
Example 3: Write the equation in standard form, state the
coordinates of the vertex, sketch the graph and state the
maximum or minimum value of the function.
f ( x) = − x2 − 2 x + 3
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3.5 Maximum and Minimum Values
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Example 4:
Find the vertex of the quadratic function and state the
maximum or minimum value. Do this by not completing the
square.
f (x ) = 2 x 2 − 12 x + 23
Example 5:
Find the vertex of the quadratic function and state the
maximum or minimum value. Do this by not completing the
square.
f (x) = − x2 + x + 2
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3.5 Maximum and Minimum Values
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Example 6:
Write the equation in standard form, state the coordinates of
the vertex, sketch the graph and state the maximum or
minimum value of the function.
f (x) = − x2 − 4 x + 2
Example 7:
Find the quadratic equation that satisfies the given conditions.
Vertex is ( –1, –8) and passes through the point ( 2, 10).
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3.5 Maximum and Minimum Values
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Example 8:
Find the quadratic equation that satisfies the given conditions.
Vertex is ( 5, 7) and passes through the point ( 3, 4).