Match the functions in (a)-(e) with the graphs in Figure 5.8...

Match the functions in (a)-(e) with the graphs in Figure 5.8 . The constants $s$ and $k$ are the same in each function.
(a) $f(x)=s$
(b) $f(x)=k x$
(c) $f(x)=k x-s$
(d) $f(x)=2 s-k x$
(e) $\quad f(x)=2 s-2 k x$

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Key Concepts

Graphical Transformations of Linear Functions

Graph transformations involve shifting, reflecting, or scaling the basic line represented by a simple equation like y = kx. By modifying constants and coefficients in the function’s algebraic form, one can determine how its graph will move vertically or horizontally, invert, or change slope, which is essential for matching equations to their corresponding graphs.

Slope and Y-Intercept

The slope of a linear function indicates how rapidly the function’s value increases or decreases as x changes, while the y-intercept represents the value of the function when x equals zero. Grasping these components helps in predicting and matching how different linear equations behave graphically.

Constant Function

A constant function always returns the same value regardless of the input, meaning its graph is a horizontal line. Understanding constant functions is key to recognizing when no change occurs as the independent variable varies.

Linear Function

A linear function is one that creates a straight-line graph and is typically expressed in slope?intercept form, y = mx + b. This concept is central to studying how changes in the slope (the multiplier of x) and the y-intercept (the constant term) influence the graph’s steepness and vertical position.

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