Question

a parabola opening up or down has vertex (0, -5) and passes through (10, 20). 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 \((0, -5)\), so \( h = 0 \) and \( k = -5 \). Substituting these into the formula, we get \( y = a(x - 0)^2 - 5 \), which simplifies to \( y = ax^2 - 5 \).

Step2: Substitute the point \((10, 20)\)

We know the parabola passes through \((10, 20)\), so substitute \( x = 10 \) and \( y = 20 \) into \( y = ax^2 - 5 \). This gives \( 20 = a(10)^2 - 5 \).

Step3: Solve for \( a \)

Simplify the equation: \( 20 = 100a - 5 \). Add 5 to both sides: \( 20 + 5 = 100a \), so \( 25 = 100a \). Then divide both sides by 100: \( a=\frac{25}{100}=\frac{1}{4} \).

Step4: Write the equation

Substitute \( a = \frac{1}{4} \), \( h = 0 \), and \( k = -5 \) back into the vertex form \( y = a(x - h)^2 + k \). We get \( y=\frac{1}{4}(x - 0)^2 - 5 \), which simplifies to \( y=\frac{1}{4}x^2 - 5 \).

Answer:

\( y = \frac{1}{4}x^2 - 5 \) (or in vertex form as \( y=\frac{1}{4}(x - 0)^2 - 5 \))