Questions: Write an equation for the parabola, in vertex form, with vertex (5,12) that passes through the point (7,20).

Solution Steps

To find the equation of a parabola in vertex form, we use the formula \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. Given the vertex \((5, 12)\), we substitute \(h = 5\) and \(k = 12\) into the equation. We then use the point \((7, 20)\) to solve for the coefficient \(a\).

The vertex form of a parabola is given by the equation:

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

where \((h, k)\) is the vertex of the parabola.

Given the vertex \((5, 12)\), we substitute \(h = 5\) and \(k = 12\) into the equation:

\[ y = a(x - 5)^2 + 12 \]

We know the parabola passes through the point \((7, 20)\). Substituting \(x = 7\) and \(y = 20\) into the equation gives:

\[ 20 = a(7 - 5)^2 + 12 \]

This simplifies to:

\[ 20 = a(2)^2 + 12 \]

\[ 20 = 4a + 12 \]

Subtracting 12 from both sides:

\[ 8 = 4a \]

Dividing by 4:

\[ a = 2 \]

Substituting \(a = 2\) back into the vertex form, we have:

\[ y = 2(x - 5)^2 + 12 \]

Final Answer