Question

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

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 \( y = -4|x + 3| + 5 \), we can rewrite \( x + 3 \) as \( x - (-3) \), so \( h=-3 \) and \( k = 5 \). Thus, the vertex of the graph is \((-3, 5)\).

Step2: Determine the direction and vertical stretch

The coefficient \( a=-4 \). Since \( a < 0 \), the graph opens downward. The absolute value of \( a \), \( | - 4|=4 \), means the graph is vertically stretched by a factor of 4.

Step3: Find two more points to plot

• When \( x=-3 + 1=-2 \), substitute into the equation: \( y=-4|-2 + 3|+5=-4|1| + 5=-4 + 5 = 1 \). So we have the point \((-2, 1)\).
• When \( x=-3-1=-4 \), substitute into the equation: \( y=-4|-4 + 3|+5=-4|-1| + 5=-4 + 5 = 1 \). So we have the point \((-4, 1)\).
• Now, plot the vertex \((-3, 5)\), and the points \((-2, 1)\) and \((-4, 1)\). Then draw a V - shaped graph (opening downward) passing through these points.

Answer:

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

1. Plot the vertex \((-3, 5)\).
2. Plot the points \((-2, 1)\) and \((-4, 1)\) (found by substituting \( x=-2 \) and \( x = - 4\) into the equation).
3. Draw a downward - opening V - shaped graph (since \( a=-4<0 \)) passing through these points, with a vertical stretch factor of 4. The graph should have its vertex at \((-3, 5)\) and pass through \((-2, 1)\) and \((-4, 1)\) (and symmetric points on the other side of the vertex).