Questions: Consider the equation (y=-(x+3)^2-3). By hand without using technology, sketch a graph of this equation on paper. Which graph A-D below most closely matches the graph you drew?

Solution Steps

The given equation is \( y = -(x+3)^2 - 3 \).

The equation \( y = -(x+3)^2 - 3 \) represents a transformation of the basic quadratic function \( y = x^2 \):

• The term \( (x+3) \) indicates a horizontal shift 3 units to the left.
• The negative sign in front of the squared term indicates a reflection over the x-axis.
• The term \(-3\) indicates a vertical shift 3 units down.

1. Start with the basic graph of \( y = x^2 \).
2. Shift the graph 3 units to the left to get \( y = (x+3)^2 \).
3. Reflect the graph over the x-axis to get \( y = -(x+3)^2 \).
4. Shift the graph 3 units down to get \( y = -(x+3)^2 - 3 \).

Final Answer