Question

graph the equation,
$y = -3\vert x + 3\vert + 5$

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

Step2: Determine the direction and vertical stretch

The coefficient \( a=-3 \). Since \( a < 0 \), the graph opens downward. The absolute value of \( a \), \( |a| = 3 \), means the graph is vertically stretched by a factor of 3.

Step3: Find two more points to plot

We can find points by substituting values of \( x \) into the equation. Let's take \( x=-3 \) (vertex), \( x=-3 + 1=-2 \), and \( x=-3-1=-4 \).

• For \( x=-2 \): \( y=-3|-2 + 3|+5=-3|1|+5=-3 + 5 = 2 \). So the point is \((-2, 2)\).
• For \( x=-4 \): \( y=-3|-4 + 3|+5=-3|-1|+5=-3 + 5 = 2 \). So the point is \((-4, 2)\).

Step4: Plot the vertex and the two points, then draw the graph

Plot the vertex \((-3, 5)\), and the points \((-2, 2)\) and \((-4, 2)\). Then draw a V - shaped graph (opening downward) passing through these points.

Answer:

The graph has a vertex at \((-3, 5)\), opens downward, is vertically stretched by a factor of 3, and passes through points like \((-2, 2)\) and \((-4, 2)\) (the actual graph is drawn by plotting these points and connecting them appropriately).