Lane Reese
01/05/2024 · Junior High School
A parabola opening up or down has vertex \( (-3,-3) \) and passes through \( \left(14, \frac{241}{16}\right) \). Write its equation in vertex form. Simplify any fractions.
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Tutor-Verified Answer
Step-by-step Solution
The vertex form of a parabola that opens up or down is given by the equation:
\[
y = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex of the parabola. In this case, the vertex is \((-3, -3)\), so we can substitute \(h = -3\) and \(k = -3\) into the equation:
\[
y = a(x + 3)^2 - 3
\]
Next, we need to find the value of \(a\). We know that the parabola passes through the point \(\left(14, \frac{241}{16}\right)\). We can substitute \(x = 14\) and \(y = \frac{241}{16}\) into the equation to solve for \(a\):
\[
\frac{241}{16} = a(14 + 3)^2 - 3
\]
Calculating \(14 + 3\):
\[
14 + 3 = 17
\]
Now we can substitute this back into the equation:
\[
\frac{241}{16} = a(17)^2 - 3
\]
Calculating \(17^2\):
\[
17^2 = 289
\]
So we have:
\[
\frac{241}{16} = 289a - 3
\]
Next, we add 3 to both sides. To do this, we convert 3 into a fraction with a denominator of 16:
\[
3 = \frac{48}{16}
\]
Thus, we have:
\[
\frac{241}{16} + \frac{48}{16} = 289a
\]
Combining the fractions:
\[
\frac{241 + 48}{16} = 289a
\]
Calculating the numerator:
\[
241 + 48 = 289
\]
So we have:
\[
\frac{289}{16} = 289a
\]
Now, we can solve for \(a\) by dividing both sides by 289:
\[
a = \frac{289}{16} \div 289 = \frac{1}{16}
\]
Now that we have \(a\), we can substitute it back into the vertex form equation:
\[
y = \frac{1}{16}(x + 3)^2 - 3
\]
Thus, the equation of the parabola in vertex form is:
\[
\boxed{y = \frac{1}{16}(x + 3)^2 - 3}
\]
Quick Answer
The equation of the parabola in vertex form is \( y = \frac{1}{16}(x + 3)^2 - 3 \).
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