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.
$$\begin{aligned}
&f(x)=\frac{1}{2} x\\
&g(x)=\frac{1}{2} x+2\\
&h(x)=4(x-2)
\end{aligned}$$

Key Concepts

Linear Functions

Linear functions are mathematical expressions that produce straight-line graphs when plotted on a Cartesian coordinate system. They are typically represented by the equation y = mx + b, where m is the slope and b is the y-intercept. Understanding linear functions is foundational for analyzing relationships with constant rates of change.

Slope-Intercept Form

The slope-intercept form, y = mx + b, is a convenient way to express linear equations. It clearly displays the slope (m), which indicates how steep the line is, and the y-intercept (b), which is the point where the line crosses the y-axis. This form simplifies both the graphing process and the understanding of the function's behavior.

Graphing on a Cartesian Coordinate System

Graphing on a rectangular (Cartesian) coordinate system involves plotting points according to their x (horizontal) and y (vertical) values and drawing the corresponding curves or lines. This fundamental skill is used to visually interpret relationships in algebra and coordinate geometry.

Transformations of Functions

Transformations involve modifications of a function's graph, such as translations (shifts), reflections, stretches, or compressions. These operations can change the position, orientation, or scale of the graph without altering its fundamental shape, enabling the analysis of how algebraic changes affect graphical representations.

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