5.1.2 Properties of Quadratic Functions in Standard Form
Key Objectives • Define, identify, and graph quadratic functions. • Identify and use maximums and minimums of quadratic functions to solve problems. Key Terms • The axis of symmetry is the line that divides the parabola into two symmetrical halves. • The standard form of a quadratic function is f ( x ) = ax 2 + bx + c , where a , b , and c are real numbers and a ≠ 0. • The minimum value of a function is the y -value of the lowest point on the graph of the function. • The maximum value of a function is the y -value of the highest point on the graph of the function. Example 1 Identifying the Axis of Symmetry Parabolas are symmetric curves. The graph of a quadratic function is a parabola where the axis of symmetry is a vertical line that passes through the parabola’s vertex. For the graph of the quadratic function f ( x ) = a ( x − h ) 2 + k , the axis of symmetry is the vertical line x = h and the vertex is the point ( h , k ).
Example 2 Graphing Quadratic Functions in Standard Form The coefficients a , b , and c in a quadratic function in standard form f ( x ) = ax 2 + bx + c can be used to determine properties of the function’s graph. • The parabola opens upward if a > 0 and downward if a < 0. • The axis of symmetry is the vertical line = x b a 2 . • The vertex is the point − − b f b , .
• The y -intercept is c .
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