In Exercises 33-38 , use completing the square to describe the graph of each function. Support y

step 1 For this problem, we need to describe and graph the function x squared minus 6x plus 12. And we

In Exercises 33-38 , use completing the square to describe...

Key Concepts

Quadratic Function

A quadratic function is a second degree polynomial, typically expressed in the form ax² + bx + c. Its graph is a parabola that opens upwards if a > 0 and downwards if a < 0. Understanding quadratic functions involves analyzing their standard form, roots, and the direction of the parabola, as well as knowing how changes in the coefficients affect the graph's shape and position.

Completing the Square

Completing the square is an algebraic technique used to transform a quadratic function from its standard form to vertex form. This process involves creating a perfect square trinomial from the quadratic and linear terms, which then allows the function to be expressed as a shifted square term plus a constant. This form makes it easier to identify key features such as the vertex, and hence, to analyze and graph the function.

Vertex Form

The vertex form of a quadratic function is given by a(x - h)² + k, where (h, k) represents the vertex of the parabola. This form makes it straightforward to see the translation of the parent function, x², and to understand the function’s maximum or minimum point, as well as the axis of symmetry. Converting to vertex form is particularly useful for graphing and analyzing the function's properties.

Graphical Representation

Graphical representation involves plotting the quadratic function and identifying its main features, such as the vertex, axis of symmetry, and direction of opening. By converting the function into vertex form through completing the square, one can easily determine these characteristics, which then inform how the parabola will appear when graphed. This visual approach helps in understanding the influence of algebraic transformations on the graph.

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