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

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

Key Concepts

Parabolic Equations

Parabolic equations represent a family of curves known as parabolas, which are symmetric and appear in many areas of mathematics and physics. Understanding these equations involves recognizing the quadratic relationship between the variables and how changes in the equation's parameters affect the shape and orientation of the parabola.

Vertex Form of a Parabola

The vertex form of a parabola is a way of expressing its equation to directly reveal the vertex, which is the turning point of the curve. This form, typically written as x = a(y - k)² + h or y = a(x - h)² + k, is useful because it makes clear the horizontal and vertical shifts from the origin and highlights the parabola's symmetry.

Completing the Square

Completing the square is an algebraic technique used to rewrite quadratic expressions into a perfect square plus a constant, which is instrumental in converting a quadratic equation from its standard form to its vertex form. This method simplifies the process of identifying the vertex and understanding the structure of the equation.

Axis of Symmetry

The axis of symmetry of a parabola is the line that bisects the curve into two mirror-image halves, passing through the vertex. In the vertex form of a parabola, the equation of the axis of symmetry can be directly read from the coordinate corresponding to the variable that has been squared, making it a key feature in the study of quadratic functions.

Graphing Parabolas

Graphing parabolas involves plotting the vertex, drawing the axis of symmetry, and determining additional points to capture the curve's shape. This process often requires understanding the transformation of the basic quadratic graph as well as the role of the coefficient that impacts the direction and width of the parabola.

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