Sketch the graph of each function by transforming the graph...

Sketch the graph of each function by transforming the graph of an appropriate function of the form $y=x^{n}$ from Figure $2 .$ Indicate all $x$ - and $y$ -intercepts on each graph.
(a) $P(x)=x^{2}-4$
(b) $Q(x)=(x-4)^{2}$
(c) $R(x)=2 x^{2}-2$
(d) $S(x)=2(x-2)^{2}$

Key Concepts

Parent Function

The parent function is the core or simplest form of a function from which other functions are derived by applying transformations. In the case of quadratic functions, the parent function is y = x², which has a standard parabolic shape. It serves as the basis for applying modifications such as shifts, stretches, and reflections that result in more complex graphs.

Horizontal Transformations

Horizontal transformations involve shifting or scaling the graph along the x-axis. Shifting the graph horizontally means replacing x with (x - h) to move the graph h units to the right (or using (x + h) for a shift to the left). These transformations adjust the position of the function without altering its overall shape.

Vertical Transformations

Vertical transformations modify the graph along the y-axis. This includes vertical shifts, where a constant is added to or subtracted from the entire function to move the graph up or down, and vertical stretching or compression, where the function is multiplied by a factor that increases or decreases its steepness. Such changes affect the height and spread of the graph without modifying the x-values of critical points.

Intercepts

Intercepts are significant points where the function crosses the coordinate axes. The x-intercepts occur when the output of the function is zero, revealing the zeros or roots of the equation, while the y-intercept is found by evaluating the function at x = 0. These intercepts help in sketching the graph and understanding the behavior of the function at specific points.

Quadratic Functions

Quadratic functions are polynomial functions of degree 2 that graph as parabolas. They are fundamental in algebra and calculus due to their predictable shape and systematic transformation properties. Recognizing how the standard quadratic form can be transformed through shifts and vertical stretches is important for analyzing and sketching their graphs efficiently.

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