In Exercises 17-34, write the quadratic function in standard form and sketch its graph. Identify

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In Exercises 17-34, write the quadratic function in standard...

Key Concepts

x-intercepts

The x-intercepts of a quadratic function are the points where the graph crosses the x-axis, which occur when f(x) = 0. Solving the quadratic equation ax² + bx + c = 0 yields the x-intercepts. These intercepts, sometimes called the roots of the equation, can be found using the quadratic formula and provide critical information regarding the behavior of the function.

Axis of Symmetry

The axis of symmetry of a quadratic function is the vertical line that passes through the vertex, given by the equation x = -b/(2a). This line divides the parabola into two mirror-image halves, making it an important tool for graphing and understanding the structure of the function.

Quadratic Function

A quadratic function is a polynomial of degree two that can be generally expressed as f(x) = ax² + bx + c. Its graph forms a parabola which either opens upward (if a > 0) or downward (if a < 0), and it is a fundamental function type in algebra that models many real-world phenomena.

Standard Form

The standard form of a quadratic function is written as f(x) = ax² + bx + c, where a, b, and c are constants. This form is useful because it directly shows the coefficients that determine the width and direction of the parabola, as well as the y-intercept, and sets the stage for further analysis such as finding the vertex and the intercepts.

Vertex

The vertex of a parabola is the point where the function reaches its maximum or minimum value. It can be found using the formula (-b/(2a), f(-b/(2a))) when the quadratic is in standard form. This point is critical in graphing the parabola as it represents its peak or lowest point and is the center of symmetry of the graph.

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