Question

a parabola opening up or down has vertex (0, 0) and passes through (8, 4). write its equation in vertex form. simplify any fractions.

Explanation:

Step1: Recall vertex form of parabola

The vertex form of a parabola opening up or down is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. Given vertex \((0, 0)\), so \( h = 0 \) and \( k = 0 \). The equation becomes \( y = a(x - 0)^2 + 0 = ax^2 \).

Step2: Substitute the point \((8, 4)\)

Substitute \( x = 8 \) and \( y = 4 \) into \( y = ax^2 \): \( 4 = a(8)^2 \).

Step3: Solve for \( a \)

Simplify \( 4 = 64a \). Divide both sides by 64: \( a=\frac{4}{64}=\frac{1}{16} \).

Step4: Write the final equation

Substitute \( a = \frac{1}{16} \) back into \( y = ax^2 \), we get \( y=\frac{1}{16}x^2 \).

Answer:

\( y = \frac{1}{16}x^2 \)