The graph of a quadratic function is given. Write the...

The graph of a quadratic function is given. Write the function's equation, selecting from the following options.
$$
\begin{array}{ll}
{f(x)=x^{2}+2 x+1} & {g(x)=x^{2}-2 x+1} \\
{h(x)=x^{2}-1} & {j(x)=-x^{2}-1}
\end{array}
$$
(Graph can't copy)

Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two that produce parabolic graphs. They are central in algebra and calculus, and their study involves understanding the effects of different coefficients on the shape, position, and orientation of the parabolic curve.

Standard Form

The standard form of a quadratic function is given by ax² + bx + c. This form is fundamental because it clearly displays the coefficients that determine key features of the graph such as its vertex, axis of symmetry, and discriminant, all of which influence the behavior of the function.

Parabola Vertex

The vertex of a parabola is its turning point—the minimum point if the graph opens upward or the maximum if it opens downward. It can be calculated using the formula (-b/(2a), f(-b/(2a))), making it a crucial concept for graphing quadratic functions and understanding their maximum or minimum values.

Parabola Orientation

The orientation of a parabola depends on the sign of the coefficient a. If a is positive, the parabola opens upward, and if a is negative, it opens downward. This concept is important because it helps determine whether the quadratic function has a maximum or a minimum, as well as how the graph will behave.

Graphical Analysis

Graphical analysis of quadratic functions involves examining the curved shape of the parabola to identify key features such as the vertex, axis of symmetry, intercepts, and direction of opening. This analysis is essential for linking an algebraic equation to its corresponding graph and for evaluating which equation best fits a given graphical representation.

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