midnight of October 01st










midnight of October 01st

Maximum and Minimum Values

Quadratic Functions

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 ex pressed 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, de termine 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 fol lowing 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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