Question

homework: graphing piecewise functions
math 3
directions: carefully graph each of the following. fill out the table to help you with graphing.

1. ( f(x) = \begin{cases} x + 5 & x < -2 \\ -2x - 1 & x geq -2 end{cases} )

tables: ( y = x + 5 ) with ( x = -2, -3, -4, -5 ) and ( y = 3, 2, 1, 0 ); ( y = -2x - 1 ) with ( x = -1, 0, 1, 2 ) and ( y = 1, -1, -3, -5 ); graph with grid

Explanation:

Step1: Analyze \( y = x + 5 \) (for \( x < -2 \))

For \( x = -3 \): \( y = -3 + 5 = 2 \)
For \( x = -4 \): \( y = -4 + 5 = 1 \)
For \( x = -5 \): \( y = -5 + 5 = 0 \)

Step2: Analyze \( y = -2x - 1 \) (for \( x \geq -2 \))

For \( x = -2 \): \( y = -2(-2) - 1 = 3 \)
For \( x = -1 \): \( y = -2(-1) - 1 = 1 \)
For \( x = 0 \): \( y = -2(0) - 1 = -1 \)
For \( x = 1 \): \( y = -2(1) - 1 = -3 \)
For \( x = 2 \): \( y = -2(2) - 1 = -5 \)

Step3: Graphing

• For \( y = x + 5 \) ( \( x < -2 \) ), plot points \((-3, 2)\), \((-4, 1)\), \((-5, 0)\) and draw a line (open circle at \( x = -2 \), since \( x < -2 \)).
• For \( y = -2x - 1 \) ( \( x \geq -2 \) ), plot points \((-2, 3)\), \((-1, 1)\), \((0, -1)\), \((1, -3)\), \((2, -5)\) and draw a line (closed circle at \( x = -2 \), since \( x \geq -2 \)).

Answer:

The table values are correct, and the graph should have two linear segments: \( y = x + 5 \) (left of \( x = -2 \), open circle at \( (-2, 3) \)) and \( y = -2x - 1 \) (right of and including \( x = -2 \), closed circle at \( (-2, 3) \)).