Make a table of values and draw graphs of the following...

Make a table of values and draw graphs of the following functions: (a) $f(x)=4$; (b) $f(x)=\frac{4 x+3}{5}$; (c) $f(x)=4 x-x^2$; (d) $f(x)=\left\{\begin{aligned} 4 & \text { if } x \geq 0 \\ -4 & \text { if } x<0\end{aligned}\right.$ $f(x)=\left\{\begin{array}{l}4 \\ -x \\ x+2\end{array}\right.$ if $x \geq 2$ if $-1<x<2$ if $x \leq-1$ (a) Form a table of values; then plot the points and connect them. The graph (Fig. 9-6) is a horizontal straight line. $$ \begin{array}{|c||c|c|c|c|} \hline x & -2 & 0 & 2 & 4 \\ \hline y & 4 & 4 & 4 & 4 \\ \hline \end{array} $$ (FIGURE CAN'T COPY) (b) Form a table of values; then plot the points and connect them. The graph (Fig. 9-7) is a straight line. $$ \begin{array}{|c||c|c|c|c|} \hline x & -2 & 0 & 2 & 4 \\ \hline y & -1 & \frac{3}{5} & \frac{11}{5} & \frac{19}{5} \\ \hline \end{array} $$ (FIGURE CAN'T COPY) (c) Form a more extensive table of values; then plot the points and connect them. The graph (Fig. 9-8) is a smooth curve. $$ \begin{array}{|c||c|c|c|c|c|c|c|} \hline x & -4 & -2 & 0 & 2 & 4 & 6 & 8 \\ \hline y & -32 & -12 & 0 & 4 & 0 & -12 & -32 \\ \hline \end{array} $$ (FIGURE CAN'T COPY) (d) Form a table of values. The graph (Fig. 9-9) is discontinuous at the point where x 0. $$ \begin{array}{|c||c|c|c|c|c|} \hline x & -4 & -2 & 0 & 2 & 4 \\ \hline y & -4 & -4 & 4 & 4 & 4 \\ \hline \end{array} $$ (FIGURE CAN'T COPY) (e) Form a table of values. The graph (Fig. 9-10) is discontinuous at the point where x 2. Note that the graph consists of three separate “pieces,” since the rule defining the function does so also. $$ \begin{array}{|r||r|r|r|r|r|r|r|} \hline x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline y & -1 & -2 & 1 & 0 & -1 & 4 & 4 \\ \hline \end{array} $$ (FIGURE CAN'T COPY)

Make a table of values and draw graphs of the following functions: (a) $f(x)=4$;
(b) $f(x)=\frac{4 x+3}{5}$;
(c) $f(x)=4 x-x^2$;
(d) $f(x)=\left\{\begin{aligned} 4 & \text { if } x \geq 0 \\ -4 & \text { if } x<0\end{aligned}\right.$
$f(x)=\left\{\begin{array}{l}4 \\ -x \\ x+2\end{array}\right.$
if $x \geq 2$
if $-1<x<2$
if $x \leq-1$
(a) Form a table of values; then plot the points and connect them. The graph (Fig. 9-6) is a horizontal straight line.
$$
\begin{array}{|c||c|c|c|c|}
\hline x & -2 & 0 & 2 & 4 \\
\hline y & 4 & 4 & 4 & 4 \\
\hline
\end{array}
$$
(FIGURE CAN'T COPY)
(b) Form a table of values; then plot the points and connect them. The graph (Fig. 9-7) is a straight line.
$$
\begin{array}{|c||c|c|c|c|}
\hline x & -2 & 0 & 2 & 4 \\
\hline y & -1 & \frac{3}{5} & \frac{11}{5} & \frac{19}{5} \\
\hline
\end{array}
$$
(FIGURE CAN'T COPY)
(c) Form a more extensive table of values; then plot the points and connect them. The graph (Fig. 9-8) is a
smooth curve.
$$
\begin{array}{|c||c|c|c|c|c|c|c|}
\hline x & -4 & -2 & 0 & 2 & 4 & 6 & 8 \\
\hline y & -32 & -12 & 0 & 4 & 0 & -12 & -32 \\
\hline
\end{array}
$$
(FIGURE CAN'T COPY)
(d) Form a table of values. The graph (Fig. 9-9) is discontinuous at the point where x 0.
$$
\begin{array}{|c||c|c|c|c|c|}
\hline x & -4 & -2 & 0 & 2 & 4 \\
\hline y & -4 & -4 & 4 & 4 & 4 \\
\hline
\end{array}
$$
(FIGURE CAN'T COPY)
(e) Form a table of values. The graph (Fig. 9-10) is discontinuous at the point where x 2. Note that the graph
consists of three separate “pieces,” since the rule defining the function does so also.
$$
\begin{array}{|r||r|r|r|r|r|r|r|}
\hline x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\
\hline y & -1 & -2 & 1 & 0 & -1 & 4 & 4 \\
\hline
\end{array}
$$
(FIGURE CAN'T COPY)

Key Concepts

Table of Values

A table of values is a method of organizing selected input (x) and output (y) pairs from a function. It is an essential tool in understanding and visualizing the behavior of functions by computing values at specific points, which can later be plotted on a coordinate system to form the graph of the function.

Graphing Functions

Graphing functions involves plotting points calculated from a function's rule on a coordinate system and then connecting those points to reveal the shape and behavior of the function. This process helps in understanding relationships, patterns, and trends within the function, and it aids in visualizing concepts such as slope, curvature, and intercepts.

Constant Function

A constant function is one where the output value remains the same for every input. This is represented by an equation of the form f(x) = c, where c is a constant. The graph of a constant function is a horizontal line, indicating that there is no variation in output regardless of the input, which is a fundamental concept in function analysis.

Linear Function

A linear function is defined by an equation in the form f(x) = mx + b, where m represents the slope and b the y-intercept. The graph of a linear function is a straight line. Understanding the slope and intercept is crucial, as these determine the rate of change and the exact position of the line on the graph.

Quadratic Function

A quadratic function has the general form f(x) = ax^2 + bx + c and its graph forms a parabola. The shape of the parabola, its vertex, axis of symmetry, and the direction it opens (upward or downward) are key properties used to analyze and sketch the graph of a quadratic function.

Piecewise-Defined Function

A piecewise-defined function is one that is described by different expressions over different parts of its domain. These functions require careful attention to the conditions under which each rule applies, as they can lead to graphs that are segmented or have abrupt changes in behavior at the boundaries between pieces.

Discontinuity

Discontinuity in a function occurs when there is a break or gap in its graph. This is common in piecewise-defined functions where the function's rule changes at certain points, resulting in jumps or holes in the graph. Understanding discontinuities is important for analyzing the overall behavior and limits of functions.

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