Question

graphing
graph the following piecewise functions on graph paper. clearly label each interval, endpoint, and key feature.
question 3:
$h(x)=\

$$\begin{cases}2x + 3 & \\text{if }x < 1 \\\\ -4x + 5 & \\text{if }1 < x < 4 \\\\ x + 2 & \\text{if }x \\geq 4\\end{cases}$$

$

Explanation:

Step1: Analyze the first piece \(h(x)=2x + 3\) for \(x<1\)

• When \(x = 0\) (a value in \(x<1\)), \(h(0)=2\times0+3=3\).
• The slope \(m = 2\), so it is a line with a positive slope.

Step2: Analyze the second piece \(h(x)=-4x + 5\) for \(1

• When \(x = 2\) (a value in \(1
• The slope \(m=-4\), so it is a line with a negative slope.
• At \(x = 1\) (not included in this piece), \(h(1)=-4\times1+5 = 1\) (but for the left - hand limit as \(x\to1^{-}\), using \(h(x)=2x + 3\), \(h(1)=2\times1+3=5\))

Step3: Analyze the third piece \(h(x)=x + 2\) for \(x\geq4\)

• When \(x = 4\), \(h(4)=4+2=6\).
• The slope \(m = 1\), so it is a line with a positive slope.

To graph:

• For \(y = 2x+3\) (\(x<1\)): Plot the point \((0,3)\) and use the slope to draw the line (open - circle at \(x = 1\) since \(x = 1\) is not included in this interval).
• For \(y=-4x + 5\) (\(1
• For \(y=x + 2\) (\(x\geq4\)): Plot the point \((4,6)\) and use the slope to draw the line (closed - circle at \(x = 4\) since \(x = 4\) is included in this interval).

Answer:

Graph the three linear pieces \(y = 2x+3\) (\(x<1\)), \(y=-4x + 5\) (\(1