Let \(L_i(t) = -2t + 1\).
Let \(L_o(t) = \frac {1}{2}t + 2\).

Let \(K(x)\) be a piecewise defined function defined by

This time, we will compose three functions: \(T = L_o \circ K \circ L_i\).

What will be domain (horizontal) effects and range (vertical) effects? The values \(L_i\) will become the inputs to \(K\). Therefore, \(L_i\) affects the domain of \(K\) and moves the graph horizontally. \(L_o\) will take the range values from \(K\) and move the graph vertically.

\(\blacktriangleright \) Domain - horizontal : \(L_i(t) = -2t + 1\)

• The leading coefficient for \(L_i\) is negative. The graph is going to be reflected horizontally.
  The parabola will be on the left and the “Vee” on the right.
• The leading coefficent is \(-2\), which speeds up the input into \(K\), which compresses the graph horizontally.
• Finally, \(L_i\) is adding \(1\), which is going into \(K\), which shifts the graph to the left.

\(\blacktriangleright \) Range - vertical : \(L_o = \frac {1}{2}t + 2\)

• The leading coefficient for \(L_o\) is \(\frac {1}{2}\). The graph is going to be compressed vertically.
  
• Finally, the graph will be shifted up \(2\).

Graph of \( y = T(m) = (L_o \circ K \circ L_i)(m)\).

Note: All of the endpoints remained the same type.

• The short arm of the “Vee” is a solid dot on both graphs.
• The long arm of the “Vee” is a hollow dot on both graphs.
• The end of the horizontal line segment nearest the “Vee” is a solid dot on both graphs.
• The end of the horizontal line segment nearest the parabola is a hollow dot on both graphs.
• The inside endpoint on the parabola is hollow on both graphs.
• The outside endpoint on the parabola is solid on both graphs.

All horizontal measurements are half what they were, so that when \(L_i\) multiplies them by \(2\), they get back to their original size for input into \(K\).

All height measurements are half what the were to begin with.

We can trace the solid left enpoint of the line segment: \((-3, K(-3)) = (-3, -6)\) on the graph of \(K\) through the transformations:

\begin{align*} \text {Start :} & \text { } & (-3, K(-3)) \\ \text {Start :} & \text { } & (-3, -6) \\ \text {Horz Shift Left $1$ :} & \text { } & (-4,-6) \\ \text {Horz Reflection :} & \text { } & (4, -6) \\ \text {Horz Compress Factor $\tfrac {1}{2}$ :} & \text { } & (2, -6) \\ \text {Vert Compress Factor $\tfrac {1}{2}$:} & \text { } & (2, -3) \\ \text {Vert Shift Up $2$ :} & \text { } & (2, -1) \\ \text {Composition :} & \text { } & (2, -1) \end{align*}

The solid endpoint of the horizontal line segment will be \((2, -1)\).