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Quadratic Functions; An Introduction Mr. J. Grossman Graph the function y = x² Describe the graph. What does it look like? Graph the function y = x² The graph is… Nonlinear U-shaped smooth curve (called a parabola) Symmetry Axis of Symmetry: the line that divides the parabola into two matching halves. It has a vertex. Graph the function y = x² The function we have graphed and described is known as a Quadratic Function. A Quadratic Function is any function that can be written in the standard form: y = ax² + bx + c where a, b, and c are real numbers and a ≠ 0. You must be able to recognize a Quadratic Function From a graph… You must be able to recognize a Quadratic Function From a table of values… x y = x^2 0 0 1st Common Difference 2nd Common Difference 1 Constant change in x-values 1 1 2 2 4 3 3 9 5 2 4 16 7 2 You must be able to recognize a Quadratic Function A quadratic function will not have a common first difference. It will have a common second difference. This is true of all quadratic functions. x y = x^2 0 0 1st Common Difference 2nd Common Difference 1 Constant change in x-values 1 1 2 2 4 3 3 9 5 2 4 16 7 2 You must be able to recognize a Quadratic Function From an equation… Must be a 2nd degree polynomial y = 2x² y = -5x² + 6x f(x) = x² + 9 f(x) = ½ x² + 3x -11 Tell whether each function is a Quadratic Function: {(-4, 8), (-2, 2), (0, 0), (2, 2), (4, 8)} y = -3x + 20 y + 3x² = -4 Quadratic Functions Some open upwards; others, downward. Quadratic Functions When a quadratic function is written in the form y = ax² + bx + c, the value of a determines the direction in which the parabola opens. If a > 0, the parabola opens upward. If a < 0, the parabola opens downward. Remember, a ≠ 0 !!! Quadratic Functions Graph the function: f(x) = -4x² - x + 1 Graph the function: y – 5x² = 2x – 6 Does the parabola open upward or downward? Does the parabola open upward or downward? Quadratic Functions The highest or lowest point on the parabola is known as the vertex. If a > 0, the parabola opens upward and the y-value of the vertex is the minimum value of the function. If a < 0, the parabola opens downward and the yvalue of the vertex is the maximum value of the function. Quadratic Functions Quadratic Functions The value of a (coefficient of the x² term) affects the width of the parabola also. The smaller the absolute value of a, the wider the parabola. Compare the graphs of: y = -4x², y = ¼ x² + 3, and y = x². Which graph (quadratic function) is widest? Narrowest? Quadratic Functions Recall that the quadratic function in standard form is written: y = ax² + bx + c. The value of c (constant of the quadratic function) translates the graph of the function up or down the axis of symmetry. Compare the graphs of the following quadratic functions: y = 2x², y = 2x² + 3, and y = 2x² - 3. Quadratic Functions Positive values of c shift the vertex UP. Negative values of c shift the vertex DOWN. Quadratic Functions DOMAIN: Unless a specific domain is given, you may assume that the domain of a quadratic function is the set of all real numbers. RANGE: The range begins (albeit minimum or maximum value) with the vertex. Any questions? If not, complete Study Guide Practice 10-1, pg 128, all problems.