Question

use the drawing tools to form the correct answer on the graph.
plot function ( h ) on the graph.
( h(x) = \begin{cases} -4, & x < -3 \\ x + 5, & x geq -3 end{cases} )
drawing tools
select
point
open point
ray
click on a tool to begin drawing.
reset
undo

Explanation:

Step1: Analyze the piecewise function for \( x < -3 \)

The function is \( h(x) = -4 \) when \( x < -3 \). This is a horizontal line. We can plot a few points: for any \( x < -3 \), say \( x = -4 \), \( h(-4) = -4 \); \( x = -5 \), \( h(-5) = -4 \). Since it's for \( x < -3 \), the point at \( x = -3 \) for this part is an open point (because \( x = -3 \) is not included here). So the open point is at \( (-3, -4) \), and we draw a horizontal line to the left from this open point.

Step2: Analyze the piecewise function for \( x \geq -3 \)

The function is \( h(x) = x + 5 \) when \( x \geq -3 \). Let's find the point at \( x = -3 \): \( h(-3) = -3 + 5 = 2 \). So this is a closed point at \( (-3, 2) \) (because \( x = -3 \) is included here). Now, find another point for \( x \geq -3 \), say \( x = 0 \), \( h(0) = 0 + 5 = 5 \), so the point is \( (0, 5) \); \( x = 1 \), \( h(1) = 1 + 5 = 6 \), point \( (1, 6) \), etc. We draw a line with slope 1 (since the equation is in slope - intercept form \( y = mx + b \) with \( m = 1 \), \( b = 5 \)) starting from the closed point \( (-3, 2) \) and going to the right.

Graphing Steps (Summary):

1. For \( x < -3 \):

• Plot an open point at \( (-3, -4) \).
• Draw a horizontal ray (line) to the left from this open point (since \( x \) values are less than - 3) with \( y=-4 \).

1. For \( x \geq -3 \):

• Plot a closed point at \( (-3, 2) \).
• Draw a ray (line) with slope 1 (going up and to the right) from this closed point using the equation \( y=x + 5 \).

(Note: Since this is a graphing problem, the final answer is the correct graph constructed as per the above steps. If we were to describe the key points: Open point at \((-3, -4)\), horizontal line left from it; closed point at \((-3, 2)\), line with slope 1 right from it.)

Answer:

The graph has an open point at \((-3, -4)\) with a horizontal line to the left, and a closed point at \((-3, 2)\) with a line \(y = x+5\) (slope 1) to the right.