# Maximum and Minimum Values

A ** quadratic function **, f, is a function that can be
put in the form,

where a, b, and care real numbers and a is not equal to zero.

A quadratic function, f, can be expressed in the form,
called the **standard
form** by the method of completing the square.

The axis of symmetry is the vertical line x = h. It divides the graph in half .

Properties of a Quadratic Function and Its Graph

Given a quadratic function in the form, with a not equal to zero .

1. The graph of f is a parabola .

2. The vertex is (h, k).

3. If a> 0, the parabola opens upward and it has a minimum value which is k.

4. If a< 0 the parabola opens downward and it has a maximum value which is k.

5. Every parabola will have a minimum or maximum value, but not both.

6. The axis of symmetry is the vertical line x = h.

Example 1: For the given quadratic function, determine if
it has a maximum or minimum value

and then find the vertex.

Example 2: For the given quadratic function, determine if
it has a maximum or minimum value

and then find the vertex.

Example 3: Determine whether the following quadratic
function has a maximum or minimum

value. Then write the quadratic function in standard form and find the maximum
or minimum

value.

Example 4: Determine whether the following quadratic
function has a maximum or minimum

value. Then write the quadratic function in standard form and find the maximum
or minimum

value.

**Finding the Vertex of a Quadratic Function **

Given a quadratic function in the form, the vertex is

Example 5: For the following quadratic function, find the vertex.

Example 6: For the following quadratic function, find the
coordinates of the maximum or

minimum point.

**Graphing Quadratic Functions with Equations in Standard
Form **

1. Determine whether the parabola opens upward or
downward.

2. Determine the vertex.

3. Find any x- intercept by replacing f(x) with 0 and then solving for x .

4. Find the y- intercept by replacing x with 0.

5. Plot the intercept (s) and vertex, sketch the graph and draw the axis of
symmetry.

Example 7: Sketch the graph of

Example 8: Sketch the graph of .

Example 9: Find a quadratic function satisfying the given
conditions.

Vertex: (-3, -1); passes through (2, -101)

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