Question
A parabola opening up or down has a vertex of (0,3) and passes through (-8,-5). Write its equation in vertex form.
(1 point)
Responses
y = -1/8x^2+3
y = -1 eighth x squared plus 3
y =1/3x^2+2
y =13x2+2
1/8x^2+3
1 eighth x squared plus 3
y=a(x−0)^2+3
(1 point)
Responses
y = -1/8x^2+3
y = -1 eighth x squared plus 3
y =1/3x^2+2
y =13x2+2
1/8x^2+3
1 eighth x squared plus 3
y=a(x−0)^2+3
Answers
GPT-4o mini
To write the equation of the parabola in vertex form, we start with the general vertex form equation:
\[ y = a(x - h)^2 + k \]
where \((h, k)\) is the vertex of the parabola. In this case, the vertex is \((0, 3)\), so the equation simplifies to:
\[ y = a(x - 0)^2 + 3 \]
\[ y = ax^2 + 3 \]
Next, we need to determine the value of \(a\). We know the parabola passes through the point \((-8, -5)\). We can substitute \(x = -8\) and \(y = -5\) into the equation to find \(a\):
\[ -5 = a(-8)^2 + 3 \]
\[ -5 = 64a + 3 \]
Now, solve for \(a\):
1. Subtract 3 from both sides:
\[-5 - 3 = 64a\]
\[-8 = 64a\]
2. Divide both sides by 64:
\[a = -\frac{8}{64} = -\frac{1}{8}\]
Now that we have \(a\), we can write the final equation of the parabola:
\[ y = -\frac{1}{8}x^2 + 3 \]
Thus, the correct response is:
**y = -1/8x^2 + 3**
\[ y = a(x - h)^2 + k \]
where \((h, k)\) is the vertex of the parabola. In this case, the vertex is \((0, 3)\), so the equation simplifies to:
\[ y = a(x - 0)^2 + 3 \]
\[ y = ax^2 + 3 \]
Next, we need to determine the value of \(a\). We know the parabola passes through the point \((-8, -5)\). We can substitute \(x = -8\) and \(y = -5\) into the equation to find \(a\):
\[ -5 = a(-8)^2 + 3 \]
\[ -5 = 64a + 3 \]
Now, solve for \(a\):
1. Subtract 3 from both sides:
\[-5 - 3 = 64a\]
\[-8 = 64a\]
2. Divide both sides by 64:
\[a = -\frac{8}{64} = -\frac{1}{8}\]
Now that we have \(a\), we can write the final equation of the parabola:
\[ y = -\frac{1}{8}x^2 + 3 \]
Thus, the correct response is:
**y = -1/8x^2 + 3**