Pregunta
Find the coordinates of the vertex and the equation of the axis of symmetry for the parabola given by the equation. \[ y=x^{2}+8 \] vertex \[ (x, y)=(\square \] axis of symmetry
Ask by Maxwell Willis. in the United States
Feb 20,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
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The vertex is at \((0, 8)\) and the axis of symmetry is \(x = 0\).
Solución
To find the coordinates of the vertex and the equation of the axis of symmetry for the parabola given by the equation
\[
y = x^2 + 8,
\]
we can follow these steps:
1. **Identify the standard form of the parabola**: The equation of a parabola in standard form is given by
\[
y = a(x - h)^2 + k,
\]
where \((h, k)\) is the vertex of the parabola.
2. **Rewrite the given equation**: The given equation can be rewritten as
\[
y = 1(x - 0)^2 + 8.
\]
From this, we can see that \(a = 1\), \(h = 0\), and \(k = 8\).
3. **Determine the vertex**: The vertex \((h, k)\) is therefore
\[
(0, 8).
\]
4. **Find the axis of symmetry**: The axis of symmetry for a parabola in this form is given by the line
\[
x = h.
\]
Since \(h = 0\), the equation of the axis of symmetry is
\[
x = 0.
\]
Now, summarizing the results:
- The coordinates of the vertex are
\[
(x, y) = (0, 8).
\]
- The equation of the axis of symmetry is
\[
x = 0.
\]
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Simplifique esta solución Extra Insights
The vertex of the parabola given by the equation \( y = x^{2} + 8 \) can be found by recognizing that this is in the standard form \( y = a(x - h)^{2} + k \). Here, \( h = 0 \) and \( k = 8 \), so the vertex is at \( (0, 8) \). The axis of symmetry for a parabola in this form is simply the vertical line that passes through the vertex. Therefore, the equation of the axis of symmetry is \( x = 0 \).
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