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 $1 .$ Indicate all $x$ - and $y$ -intercepts on each graph.
a. $P(x)=x^{4}-16$
b. $P(x)=-(x+5)^{4}$
c. $P(x)=-5 x^{4}+5$
d. $P(x)=(x-5)^{4}$

Key Concepts

Graph Transformations

Graph transformations refer to operations applied to a basic function to produce a new function. These operations include translations (shifting the graph horizontally or vertically), reflections (flipping the graph over the x-axis or y-axis), and dilations (stretching or compressing the graph vertically or horizontally). In the context of the problem, the functions are obtained by performing these transformations on the base function y = x?.

Intercepts

Intercepts are the points where a graph crosses the coordinate axes: the x-intercepts are where the graph crosses the x-axis (y = 0) and the y-intercepts where it crosses the y-axis (x = 0). Finding these intercepts typically involves setting the appropriate variable to zero and solving the resulting equation, which helps in sketching the overall graph.

Even-Degree Polynomial Functions

Even-degree polynomial functions, such as those involving x?, generally have graphs that are symmetric about a vertical line. The basic graph of y = x? has a distinctive U-shaped curve that is influenced by any transformations applied to it. The end behavior of these functions is determined by the leading coefficient, which indicates whether both ends of the graph rise or fall.

Horizontal and Vertical Shifts

Horizontal shifts occur when x is replaced by (x + c) or (x - c), moving the graph left or right, respectively. Vertical shifts result from adding or subtracting a constant from the entire function, moving the graph up or down. These shifts are used to reposition the vertex or key points of the graph, aiding in the accurate sketching of the transformed functions.

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