Quadratic Functions, Quadratic Expressions, Quadratic Equations

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Presentation transcript:

Quadratic Functions, Quadratic Expressions, Quadratic Equations Definition: A quadratic function is a function of the form where a, b, c are real numbers and a  0. The expression on the right-hand-side is call a quadratic expression.

Quadratic Expressions: Factored Form Examples: 1. 2.

Factoring quadratic expressions: Given where a, b, c are integers. Case 1: a = 1; Since

we have

Examples:

Case 2: where a, b, c are integers and a  1. Since

we have

Examples:

Quadratic Equations: A quadratic equation is an equation of the form: Problem: Find the real numbers x, if any, that satisfy the equation. The numbers that satisfy the equation are called solutions or roots.

Methods of Solution: Method 1: Factor then the solutions (roots) of the equation are

Examples:

Method 2: Use the QUADRATIC FORMULA The real number solutions (roots) of the quadratic equation are: provided Method 2: Use the QUADRATIC FORMULA

The quadratic formula is often written as The number is called the discriminant.

The Discriminant: Given the quadratic equation If:

(1) ; the roots are: (2) the roots are: (3) no real roots.

Examples:

Quadratic Functions: The graph of is a parabola. The graph looks like if a > 0 if a < 0

Key features of the graph: The maximum or minimum point on the graph is called the vertex. The x-coordinate of the vertex is:

The y-intercept; the y-coordinate of the point where the graph intersects the y-axis. The y-intercept is: The x-intercepts; the x-coordinates of the points, if any, where the graph intersects the x-axis. To find the x-intercepts, solve the quadratic equation

Examples: Sketch the graph of vertex: y-intercept: x-intercepts:

Sketch the graph of Vertex: y-intercept: x-intercept(s):

Sketch the graph of Vertex: y-intercept: x-intercept(s):