# COMPYS1314 4 1 Notes quadratic fcns by Dk14f7H

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Definition: A quadratic function is a function which can be written in the form f ( x)  ax 2  bx  c
where a  0 .
The graph of a quadratic function is a parabola.

Definition: The maximum value of a function is the largest y-value on the graph. The minimum
value of a function is the smallest y-value on the graph.

Standard form for a quadratic function:
Every quadratic function f ( x)  ax 2  bx  c ,where a  0 , can be written in the form
f ( x )  a ( x  h) 2  k .
This is called standard form for a quadratic function. To write a quadratic function
f ( x)  ax 2  bx  c in standard form, we need to complete the square which results the following
b
formulation of h                and k  f (h) .
2a
b
 The vertex of the parabola is (h, k ) where h      and k  f (h) .
2a
 The parabola is symmetric with respect to the vertical line x  h . This line is
called axis of symmetry for the parabola.

   If a  0 , the graph of f ( x) opens down and f f ( x) has a maximum value.

   If a  0 , the graph of f ( x) opens up and f ( x) has a minimum value.

   The larger a , is the narrower the parabola is.

Example 1: Sketch the graph of f ( x)  5  4 x  x 2 . State the vertex and intercepts. What is the
maximum or minimum value and where they are attained? Find the domain and range.
4.1.2

Example 2: Sketch the graph of f ( x)  2 x 2  12 x  16 . State the vertex and intercepts. What is the
maximum or minimum value and where they are attained? Find the domain and range.

Example 3: Sketch the graph of f ( x)  3x 2  15 x  18 . State the vertex and intercepts. What is the
maximum or minimum value and where they are attained? Find the domain and range.
4.1.3

Example 4: Sketch the graph of f ( x)  2 x 2  16 x  29 . State the vertex and intercepts. What is the
maximum or minimum value and where they are attained? Find the domain and range.

Example 5: Sketch the graph of f ( x)  2 x 2  12 x  16 . State the vertex and intercepts. What is the
maximum or minimum value and where they are attained? Find the domain and range.
4.1.4

Example 6: Write f ( x)  x 2  x  2 in standard form. What is its maximum or minimum
value? What input produces this maximum or minimum?

Example 7:   Find the maximum or minimum value of f ( x)  2 x 2  5 x  1 .

Example 8:   Find the quadratic function such that f (3)  6 and the vertex is (2, 3) .

HOMEWORK 4.1:
4.1                    Pg. 265: 24, 28,32, 36-50 EVEN ,54-62 EVEN , 76

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