Sketch the graph of each quadratic function and compare it...

Sketch the graph of each quadratic function and compare it with the graph of $y=x^{2}$.
(a) $f(x)=-\frac{1}{2}(x-2)^{2}+1$
(b) $g(x)=\left[\frac{1}{2}(x-1)\right]^{2}-3$
(c) $h(x)=-\frac{1}{2}(x+2)^{2}-1$
(d) $k(x)=[2(x+1)]^{2}+4$

Key Concepts

Quadratic Functions

A quadratic function is a polynomial function of degree two and is generally represented by the standard form f(x) = ax² + bx + c. The graph of a quadratic function is a parabola, which opens upward if a > 0 and downward if a < 0. This basic form helps to understand how different parameters affect the shape and position of the parabola.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. This form clearly shows the horizontal and vertical translations needed from the parent function y = x² and simplifies the process of graphing by identifying the vertex directly and indicating the direction of opening and the width of the parabola.

Graphical Transformations

Graphical transformations include shifting, reflecting, stretching, and compressing the graph of the basic function y = x². These transformations allow you to modify the position and shape of the parabola. Horizontal shifts occur when the x-variable is replaced by (x - h), vertical shifts occur when k is added or subtracted, reflections are due to a negative coefficient a, and stretching or compressing happens depending on the absolute value of a being greater or less than 1.

Vertical Reflection, Stretching, and Compression

The coefficient a in the vertex form not only determines if the parabola opens up or down (with a negative value indicating a reflection over the x-axis) but also affects the vertical scaling of the parabola. A value of |a| greater than 1 compresses the graph vertically, making it narrower, while a value between 0 and 1 stretches the graph vertically, making it wider compared to the base graph y = x².

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