Question

1. what is the equation written in vertex form of a parabola with a vertex of (-5,8) that passes through the point (-3,4).

Explanation:

Step1: Recall Vertex Form

The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. Given vertex \((-5, 8)\), so \( h = -5 \), \( k = 8 \). Substitute into the formula: \( y = a(x - (-5))^2 + 8 = a(x + 5)^2 + 8 \).

Step2: Find the Value of \( a \)

The parabola passes through \((-3, 4)\), so substitute \( x = -3 \), \( y = 4 \) into the equation:
\( 4 = a(-3 + 5)^2 + 8 \)
Simplify inside the parentheses: \( -3 + 5 = 2 \)
Then square it: \( 2^2 = 4 \)
So the equation becomes \( 4 = a(4) + 8 \)
Subtract 8 from both sides: \( 4 - 8 = 4a \)
\( -4 = 4a \)
Divide both sides by 4: \( a = -1 \)

Step3: Write the Final Equation

Substitute \( a = -1 \), \( h = -5 \), \( k = 8 \) back into the vertex form:
\( y = -1(x + 5)^2 + 8 \) or \( y = -(x + 5)^2 + 8 \)

Answer:

\( y = -(x + 5)^2 + 8 \)