To find the equation of a parabola in standard form, we can use the vertex form of a quadratic equation, which is given by:
\[ y = a(x - h)^2 + k \]
Where \( (h, k) \) is the vertex of the parabola. Given the vertex is \( (3, -1) \), we have:
\[ y = a(x - 3)^2 - 1 \]
Next, since we know that the parabola opens upwards, the value of \( a \) will be positive. We also know that the parabola crosses the y-axis at \( y = 8 \). The y-intercept occurs when \( x = 0 \), so we can substitute \( x = 0 \) and \( y = 8 \) into the equation to find \( a \):
\[ 8 = a(0 - 3)^2 - 1 \] \[ 8 = a(9) - 1 \] \[ 8 + 1 = 9a \] \[ 9 = 9a \] \[ a = 1 \]
Now that we have determined that \( a = 1 \), we can write the equation of the parabola:
\[ y = (x - 3)^2 - 1 \]
To convert this to standard form \( ax^2 + bx + c \), we can expand the equation:
\[ y = (x - 3)(x - 3) - 1 \] \[ y = x^2 - 6x + 9 - 1 \] \[ y = x^2 - 6x + 8 \]
Therefore, the equation of the parabola in standard form is:
\[ y = x^2 - 6x + 8 \]
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