Question

a parabola opening up or down has vertex (1, 0) and passes through \\(\left(-5, \frac{9}{2}\
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 \((1,0)\), so \(h = 1\) and \(k = 0\). Substituting these into the formula, we get \( y = a(x - 1)^2+0=a(x - 1)^2 \).

Step2: Substitute the point \((-5,\frac{9}{2})\)

We know the parabola passes through \((-5,\frac{9}{2})\), so substitute \(x=-5\) and \(y = \frac{9}{2}\) into \( y = a(x - 1)^2 \).

$$ \frac{9}{2}=a(-5 - 1)^2 $$

Simplify \((-5 - 1)^2=(-6)^2 = 36\), so the equation becomes:

$$ \frac{9}{2}=a\times36 $$

Step3: Solve for \(a\)

To find \(a\), divide both sides by 36:

$$ a=\frac{\frac{9}{2}}{36}=\frac{9}{2}\times\frac{1}{36}=\frac{1}{8} $$

Step4: Write the equation

Substitute \(a=\frac{1}{8}\), \(h = 1\) and \(k = 0\) back into the vertex form \(y=a(x - h)^2 + k\). We get:

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

Answer:

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