Question

a parabola opening up or down has vertex (5, 3) and passes through (4, 25/8). 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. Here, \( h = 5 \) and \( k = 3 \), so the equation becomes \( y = a(x - 5)^2 + 3 \).

Step2: Substitute the point \((4, \frac{25}{8})\) into the equation

Substitute \( x = 4 \) and \( y = \frac{25}{8} \) into \( y = a(x - 5)^2 + 3 \):

$$ \frac{25}{8}=a(4 - 5)^2+3 $$

Simplify \( (4 - 5)^2 = (-1)^2 = 1 \), so the equation is \( \frac{25}{8}=a(1)+3 \).

Step3: Solve for \( a \)

Subtract 3 from both sides: \( \frac{25}{8}-3=a \). Convert 3 to eighths: \( 3=\frac{24}{8} \), so \( \frac{25}{8}-\frac{24}{8}=a \), which gives \( a=\frac{1}{8} \).

Step4: Write the final equation

Substitute \( a = \frac{1}{8} \), \( h = 5 \), and \( k = 3 \) back into the vertex form: \( y=\frac{1}{8}(x - 5)^2+3 \).

Answer:

\( y=\frac{1}{8}(x - 5)^2+3 \)