Answer :
To determine the direction in which the parabola defined by the equation \((y+3)^2 = -16(x-8)\) opens, we can follow these steps:
1. Identify the standard form of the parabola equation:
- The equation \((y+3)^2 = -16(x-8)\) is of the form \((y - k)^2 = 4p(x - h)\).
- Here, \((h, k)\) is the vertex of the parabola.
- The sign of \(4p\) determines the direction in which the parabola opens.
2. Compare the given equation to the standard form:
- Rewrite the given equation in a comparative form: \((y - (-3))^2 = -16(x - 8)\).
- This shows that \(h = 8\) and \(k = -3\), and \(4p = -16\).
3. Determine the value of \(p\):
- To find \(p\), we solve the equation \(4p = -16\):
[tex]\[ 4p = -16 \implies p = \frac{-16}{4} \implies p = -4 \][/tex]
4. Interpret the value of \(p\):
- If \(p\) is positive, the parabola opens to the right.
- If \(p\) is negative, the parabola opens to the left.
Since \(p\) is \(-4\), which is negative, the parabola opens to the left.
Answer: opens left
1. Identify the standard form of the parabola equation:
- The equation \((y+3)^2 = -16(x-8)\) is of the form \((y - k)^2 = 4p(x - h)\).
- Here, \((h, k)\) is the vertex of the parabola.
- The sign of \(4p\) determines the direction in which the parabola opens.
2. Compare the given equation to the standard form:
- Rewrite the given equation in a comparative form: \((y - (-3))^2 = -16(x - 8)\).
- This shows that \(h = 8\) and \(k = -3\), and \(4p = -16\).
3. Determine the value of \(p\):
- To find \(p\), we solve the equation \(4p = -16\):
[tex]\[ 4p = -16 \implies p = \frac{-16}{4} \implies p = -4 \][/tex]
4. Interpret the value of \(p\):
- If \(p\) is positive, the parabola opens to the right.
- If \(p\) is negative, the parabola opens to the left.
Since \(p\) is \(-4\), which is negative, the parabola opens to the left.
Answer: opens left
