Question

a parabola opening up or down has vertex (1, 7) and passes through \\(\left(-2, \frac{47}{8}\
ight)\\). 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 \((1, 7)\), so \( h = 1 \), \( k = 7 \). Substitute into the formula: \( y = a(x - 1)^2 + 7 \).

Step2: Substitute the point \((-2, \frac{47}{8})\)

We know the parabola passes through \((-2, \frac{47}{8})\), so substitute \( x = -2 \), \( y = \frac{47}{8} \) into \( y = a(x - 1)^2 + 7 \):

$$ \frac{47}{8}=a(-2 - 1)^2 + 7 $$

Step3: Solve for \( a \)

Simplify the equation:
First, calculate \((-2 - 1)^2 = (-3)^2 = 9\). So the equation becomes:

$$ \frac{47}{8}=9a + 7 $$

Subtract 7 from both sides. 7 is \(\frac{56}{8}\), so:

$$ \frac{47}{8}-\frac{56}{8}=9a $$

$$ \frac{47 - 56}{8}=9a $$

$$ \frac{-9}{8}=9a $$

Divide both sides by 9:

$$ a=\frac{-9}{8}\div9=\frac{-9}{8}\times\frac{1}{9}=\frac{-1}{8} $$

Step4: Write the final equation

Substitute \( a = -\frac{1}{8} \), \( h = 1 \), \( k = 7 \) into the vertex form:

$$ y = -\frac{1}{8}(x - 1)^2 + 7 $$

Answer:

\( y = -\dfrac{1}{8}(x - 1)^2 + 7 \)