Use the equation of the parabola in standard form...

Use the equation of the parabola in standard form $y=a(x-h)^{2}+k$ to determine the coordinates of the vertex and the equation of the axis of symmetry (complete the square if necessary). Then graph the parabola.
$$y=-x^{2}-3 x+2$$
CAN'T COPY THE GRAPH

Key Concepts

Vertex of a Parabola

The vertex of a parabola is the point where the curve attains its maximum or minimum value, depending on whether it opens downward or upward. In vertex form, the vertex is easily identified as (h, k), and the vertex is critical in understanding the parabola's position relative to the coordinate axes.

Axis of Symmetry

The axis of symmetry of a parabola is a vertical line that passes through the vertex and divides the parabola into two mirror-image halves. In vertex form, its equation can be directly read as x = h. This concept is important as it helps in accurately sketching the parabola and analyzing its geometric properties.

Graphing Quadratic Functions

Graphing a quadratic function involves plotting its vertex, axis of symmetry, and other key points such as the y-intercept. Transformations such as vertical stretching/compressing and reflecting over the x-axis, which are determined by the coefficient 'a', play a critical role in shaping the graph of the function.

Completing the Square

Completing the square is a method used to rewrite a quadratic expression from standard form into vertex form. This technique involves creating a perfect square trinomial from the quadratic and linear terms, which then reveals the vertex of the parabola. It is a fundamental algebraic tool for transforming quadratic functions.

Quadratic Function Standard Form

A quadratic function is typically written as y = ax^2 + bx + c. This form clearly shows the coefficients that determine the shape and position of the parabola, but it does not immediately reveal key features like the vertex or axis of symmetry.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. This form is useful because it directly provides the vertex, making it easier to graph the function and understand its transformations from the basic quadratic y = x^2.

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