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Quadratic Functions, Quadratic Expressions, Quadratic Equations

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.

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Quadratic Functions, Quadratic Expressions, Quadratic Equations

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  1. 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.

  2. Quadratic Expressions: Factored Form Examples: 1. 2.

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

  4. we have

  5. Examples:

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

  7. we have

  8. Examples:

  9. 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.

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

  11. Examples:

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

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

  14. The Discriminant: Given the quadratic equation If:

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

  16. Examples:

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

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

  19. 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

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

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

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

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