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Published byWillis Elliott Modified over 5 years ago

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**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.

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**Quadratic Expressions: Factored Form**

Examples: 1. 2.

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**Factoring quadratic expressions:**

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

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we have

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Examples:

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

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we have

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Examples:

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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.

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

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Examples:

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**Method 2: Use the QUADRATIC FORMULA**

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

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**The quadratic formula is often written as**

The number is called the discriminant.

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The Discriminant: Given the quadratic equation If:

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

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Examples:

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

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**Key features of the graph:**

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

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**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

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

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

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

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