Question

graph the equation.
$y = 3|x + 5| + 4$

Explanation:

Step1: Identify the vertex form of absolute value function

The general form of an absolute value function is \( y = a|x - h| + k \), where \((h, k)\) is the vertex. For the given equation \( y = 3|x + 5| + 4 \), we can rewrite \( x + 5 \) as \( x - (-5) \), so \( h=-5 \) and \( k = 4 \). Thus, the vertex of the graph is at \((-5, 4)\).

Step2: Determine the direction and stretch

The coefficient \( a = 3 \), which is positive, so the graph opens upwards. The absolute value of \( a \) (3) is greater than 1, so the graph is vertically stretched by a factor of 3 compared to the parent function \( y = |x| \).

Step3: Find additional points

To graph the function, we can find a few more points. Let's choose \( x=-4 \):
Substitute \( x=-4 \) into the equation: \( y = 3|-4 + 5| + 4 = 3|1| + 4 = 3 + 4 = 7 \). So the point \((-4, 7)\) is on the graph.
Now choose \( x=-6 \):
Substitute \( x=-6 \) into the equation: \( y = 3|-6 + 5| + 4 = 3|-1| + 4 = 3 + 4 = 7 \). So the point \((-6, 7)\) is on the graph.
We can also use \( x = 0 \):
Substitute \( x = 0 \) into the equation: \( y = 3|0 + 5| + 4 = 3\times5 + 4 = 15 + 4 = 19 \), but since the graph is a V - shape with vertex at \((-5,4)\) and opening upwards, we can plot the vertex \((-5,4)\), and the points \((-4,7)\) and \((-6,7)\), then draw the two rays of the absolute - value graph (the left ray going from the vertex through \((-6,7)\) and the right ray going from the vertex through \((-4,7)\) and extending further).

(Note: Since the question is about graphing, the final answer is the graph with vertex at \((-5,4)\), opening upwards, vertically stretched by a factor of 3, and passing through points like \((-4,7)\) and \((-6,7)\) etc. But as we can't draw the graph here, we describe the steps to graph it.)

Answer:

To graph \( y = 3|x + 5| + 4 \):

1. Plot the vertex at \((-5, 4)\).
2. Since \( a = 3>0 \), the graph opens upward and is vertically stretched by a factor of 3.
3. Plot additional points (e.g., \((-4, 7)\), \((-6, 7)\)) and draw the V - shaped graph with the vertex at \((-5, 4)\) and the two arms passing through the plotted points.