Question

1. the vertex of a quadratic is (3,-4) and it passes through the point (5,4)

Explanation:

Step1: Recall vertex form of quadratic

The vertex form of a quadratic function is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. Given vertex \((3, -4)\), so \( h = 3 \), \( k = -4 \). Substitute into the formula: \( y = a(x - 3)^2 - 4 \).

Step2: Substitute the point \((5, 4)\)

We know the graph passes through \((5, 4)\), so substitute \( x = 5 \), \( y = 4 \) into \( y = a(x - 3)^2 - 4 \):
\( 4 = a(5 - 3)^2 - 4 \)
Simplify: \( 4 = a(2)^2 - 4 \) → \( 4 = 4a - 4 \).
Add 4 to both sides: \( 8 = 4a \).
Divide by 4: \( a = 2 \).

Step3: Write the final equation

Substitute \( a = 2 \), \( h = 3 \), \( k = -4 \) back into vertex form:
\( y = 2(x - 3)^2 - 4 \). If we expand it:
\( y = 2(x^2 - 6x + 9) - 4 = 2x^2 - 12x + 18 - 4 = 2x^2 - 12x + 14 \).

Answer:

The quadratic function is \( y = 2(x - 3)^2 - 4 \) (or expanded form \( y = 2x^2 - 12x + 14 \))