Question

a parabola opening up or down has vertex (-1, -1) and passes through ( left(8, \frac{77}{4}
ight) ). write its equation in vertex form. simplify any fractions.

Explanation:

Step1: Recall vertex form of parabola

The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \((h,k)\) is the vertex. Given vertex \((-1,-1)\), so \( h=-1 \), \( k = -1 \). Substitute into the formula: \( y = a(x - (-1))^2 + (-1)=a(x + 1)^2-1 \).

Step2: Substitute the point \((8,\frac{77}{4})\)

We know the parabola passes through \((8,\frac{77}{4})\), so substitute \( x = 8 \), \( y=\frac{77}{4} \) into \( y = a(x + 1)^2-1 \):

$$ \frac{77}{4}=a(8 + 1)^2-1 $$

Simplify the right - hand side: \( 8 + 1=9 \), so \( 9^2 = 81 \), the equation becomes \( \frac{77}{4}=81a-1 \).

Step3: Solve for \( a \)

Add 1 to both sides of the equation. We know that \( 1=\frac{4}{4} \), so \( \frac{77}{4}+\frac{4}{4}=\frac{77 + 4}{4}=\frac{81}{4} \). The equation is now \( \frac{81}{4}=81a \).
Divide both sides by 81: \( a=\frac{\frac{81}{4}}{81}=\frac{81}{4}\times\frac{1}{81}=\frac{1}{4} \).

Step4: Write the vertex form equation

Substitute \( a=\frac{1}{4} \), \( h=-1 \), \( k = -1 \) into the vertex form \( y = a(x - h)^2 + k \):
\( y=\frac{1}{4}(x + 1)^2-1 \)

Answer:

\( y=\frac{1}{4}(x + 1)^2-1 \)