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Understanding Quadratic Functions: Properties, Graphs, and Vertex Form

Explore the world of quadratic functions defined by ( f(x) = ax^2 + bx + c ) where ( a <br>eq 0 ). Discover how their graphs form parabolas that can open upwards or downwards based on the coefficient ( a ). Learn about key features such as the vertex, which represents the minimum or maximum point, and the axis of symmetry. Understand the significance of the discriminant for determining the nature of x-intercepts, and how to convert to standard form using the technique of completing the square.

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Understanding Quadratic Functions: Properties, Graphs, and Vertex Form

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  1. Section 9D Quadratic Functions and Their Properties

  2. QUADRATIC FUNCTIONS A quadratic function of x is a function that can be represented by an equation of the form f (x) = ax2 + bx + c where a, b, and c are real numbers and a ≠ 0. The domain of a quadratic function is all real numbers.

  3. GRAPHS OF QUADRATIC FUNCTIONS • The graph of a quadratic function is a parabola. • The parabola opens up if the coefficient of x2 is positive. • The parabola opens down if the coefficient of x2 is negative.

  4. GRAPHS OF QUADRATIC FUNCTIONS • The vertex of a parabola is the lowest point on a parabola that opens up or the highest point on a parabola that opens down. • The axis of symmetry is the vertical line passing through the vertex of a parabola.

  5. STANDARD FORM OF QUADRATIC FUNCTIONS Every quadratic function given by f (x) =ax2+bx+ c can be written in the standard form of a quadratic function: f (x) = a(x − h)2 + k, a ≠ 0 The graph of f is a parabola with vertex (h, k). The parabola opens up if a is positive, and it opens down if a is negative. To find the standard form of a quadratic function, use the technique of completing the square.

  6. VERTEX FORMULA The vertex of the graph of f (x)  = ax2 + bx + c is

  7. SUMMARY OF PROPERTIES OF THE GRAPH OF A QUADRATIC FUNCTION • Vertex = • Axis of Symmetry: the line x = −b/(2a) • Parabola opens up if is a > 0; the vertex is a minimum point. • Parabola opens down if is a < 0; the vertex is a maximum point. f (x)  = ax2 + bx + c, a≠ 0

  8. x-INTERCEPTS OF A QUADRATIC FUNCTION • If the discriminant b2− 4ac > 0, then graph of f (x) = ax2 + bx + c has two distinct x-intercepts so it crosses the x-axis in two places. • If the discriminant b2− 4ac = 0, then graph of f (x) = ax2 + bx + c has one x-intercept so it touches the x-axis in at its vertex. • If the discriminant b2− 4ac < 0, then graph of f (x) = ax2 + bx + c has no x-intercept so it does not cross or touch the x-axis.

  9. MAXIMUM OR MINIMUM VALUE OF A QUADRATIC FUNCTION • If a is positive, then the vertex (h, k) is the lowest point on the graph of f (x)  = a(x − h)2 + k, and the y-coordinate k of the vertex is the minimum value of the function f. • If a is negative, then the vertex (h, k) is the highest point on the graph of f (x)  = a(x − h)2 + k, and the y-coordinate k of the vertex is the maximum value of the function f. • In either case, the maximum or minimum value is achieved when x = h.

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