Sketch the graphs of the three functions by hand on the same...

Sketch the graphs of the three functions by hand on the same rectangular coordinate system. Verify your results with a graphing utility.
$$\begin{aligned}
&f(x)=x^{2}\\
&g(x)=x^{2}+2\\
&h(x)=(x-2)^{2}
\end{aligned}$$

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Key Concepts

Vertical Shift

A vertical shift is a type of graph transformation where a function's graph is moved up or down on the coordinate plane. By adding a constant term to the function, such as in the transformation from f(x)=x^2 to g(x)=x^2+2, the graph is raised by the constant number of units.

Horizontal Shift

A horizontal shift is a graph transformation that moves the graph left or right along the x-axis. This is achieved by replacing the variable x with (x - c), where c represents the number of units to shift. For instance, h(x) = (x-2)^2 shifts the graph of the basic quadratic function to the right by 2 units.

Graph Transformations

Graph transformations involve altering the position, shape, or size of a graph through operations such as translations, reflections, and dilations. In the context of quadratic functions, transformations help in understanding how the graph changes when constants are added or subtracted from the equation.

Quadratic Functions

Quadratic functions are polynomial functions of degree two and are generally represented by parabolic graphs. They have key characteristics including a vertex, axis of symmetry, and a predictable end behavior, which make them central in algebra and calculus.

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