vertex (−5, −7), vertical axis of symmetry, through the point P(7, 11)

To find the equation of a parabola with the following characteristics:

- Vertex at \( (-5, -7) \)
- Vertical axis of symmetry
- Passes through the point \( P(7, 11) \)

We will use the vertex form of the equation of a parabola:

\[
y = a(x - h)^2 + k
\]

Where:
- \( (h, k) \) is the vertex, which in this case is \( (-5, -7) \),
- \( a \) is the parameter that determines the width and direction of the parabola.

Step 1: Substitute the vertex \((-5, -7)\) into the equation:

\[
y = a(x + 5)^2 - 7
\]

Step 2: Use the point \( P(7, 11) \) to find \( a \). Substitute \( x = 7 \) and \( y = 11 \):

\[
11 = a(7 + 5)^2 - 7
\]

Simplify:

\[
11 = a(12)^2 - 7
\]
\[
11 = 144a - 7
\]
\[
18 = 144a
\]
\[
a = \frac{1}{8}
\]

Step 3: Substitute \( a = \frac{1}{8} \) back into the equation:

\[
y = \frac{1}{8}(x + 5)^2 - 7
\]

Thus, the equation of the parabola is:

\[
y = \frac{1}{8}(x + 5)^2 - 7
\]

Would you like more details or further explanation?

Here are 5 related questions you might find useful:

1. How do you determine the axis of symmetry for a parabola?
2. What does the parameter \(a\) control in the equation of a parabola?
3. How do you convert a parabola from vertex form to standard form?
4. What would the equation look like if the axis of symmetry were horizontal?
5. How do you find the focus and directrix of a parabola?

**Tip:** The sign of \(a\) in the parabola's equation determines if it opens upwards (\(a > 0\)) or downwards (\(a < 0\)).

vertex (-5, -7), vertical axis of symmetry, through the point P(7, 11)

Discover how to find the equation of a parabola with a vertex at (-5, -7) and a vertical axis of symmetry passing through the point P(7, 11). This detailed solution uses the vertex form of a parabola and explains each step for students in Grades 9-12.

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