Question

graph the equation shown below by transforming the given graph of the parent function.
$y = -x^2 - 4$

Explanation:

Step1: Identify Parent Function

The parent function here is \( y = x^2 \), which is a parabola opening upwards with vertex at \((0,0)\).

Step2: Analyze Transformations

• Reflection: The negative sign in front of \( x^2 \) (\( y = -x^2 - 4 \)) reflects the parent function \( y = x^2 \) over the \( x \)-axis. So the parabola will now open downwards.
• Vertical Shift: The \(-4\) at the end indicates a vertical shift. Since it's subtracted, we shift the graph of \( y = -x^2 \) down by 4 units.

Step3: Determine Vertex and Key Points

• For the parent function \( y = x^2 \), key points are \((-2,4)\), \((-1,1)\), \((0,0)\), \((1,1)\), \((2,4)\).
• After reflection (\( y = -x^2 \)), these points become \((-2,-4)\), \((-1,-1)\), \((0,0)\), \((1,-1)\), \((2,-4)\).
• After vertical shift down 4 units (\( y = -x^2 - 4 \)), the points become \((-2,-8)\), \((-1,-5)\), \((0,-4)\), \((1,-5)\), \((2,-8)\). The vertex is now at \((0, -4)\) and the parabola opens downwards.

To graph \( y=-x^2 - 4 \):

1. Start with the parent parabola \( y = x^2 \).
2. Reflect it over the \( x \)-axis (so it opens downward).
3. Shift the entire graph down by 4 units. The vertex moves from \((0,0)\) to \((0, -4)\), and other points follow the transformation rules as calculated above.

Answer:

To graph \( y = -x^2 - 4 \):

1. Reflect the parent function \( y = x^2 \) (upward - opening parabola) over the \( x \)-axis (now opens downward).
2. Shift the reflected graph (\( y = -x^2 \)) down 4 units. The vertex is at \((0, -4)\), and the parabola opens downward, with key points like \((-2, -8)\), \((-1, -5)\), \((0, -4)\), \((1, -5)\), \((2, -8)\).