In Exercises 13–16, use the vertex and intercepts to sketch the graph of each quadratic function

step 1 Okay, we want to sketch this function and find its domain and range. And the highest coefficient

In Exercises 13–16, use the vertex and intercepts to sketch...

Key Concepts

Quadratic Function

A quadratic function is a second-degree polynomial function of the form f(x) = ax² + bx + c, where a, b, and c are constants and a is nonzero. Its graph is a parabola that opens upward when a is positive and downward when a is negative. Understanding its structure is fundamental to exploring various attributes like vertex, axis of symmetry, and intercepts.

Vertex

The vertex of a parabola is the point where the function attains its maximum or minimum value, depending on whether the parabola opens downward or upward, respectively. It represents the point of symmetry for the quadratic function and is important for graphing because it provides a reference for the shape and position of the parabola on the coordinate plane.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two mirror-image halves. It is given by the formula x = -b/(2a) when the quadratic function is in standard form. This concept is crucial for graphing because it helps in reflecting points on the parabola to complete the graph.

Intercepts

Intercepts are the points at which the graph of the function crosses the coordinate axes. The y-intercept is found by evaluating the function at x = 0, while the x-intercepts (or roots) are the solutions to the equation f(x) = 0. These intercepts provide valuable points that are used to accurately sketch the quadratic function.

Domain and Range

The domain of a quadratic function typically includes all real numbers due to the nature of polynomials. The range, however, is determined by the vertex and the direction in which the parabola opens; it includes all y-values above or below the vertex's y-coordinate. This analysis is essential for understanding the behavior and limitations of the function.

Graph Sketching

Graph sketching for quadratic functions involves plotting key points such as the vertex, intercepts, and additional symmetric points, followed by drawing the symmetrical parabola. This visual approach helps in understanding the overall behavior of the function and verifying algebraic solutions, making it a practical application of the concepts of vertex, axis of symmetry, and intercepts.

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