To find the equation of the parabola in vertex form, we can use the given vertex and a point on the parabola. The vertex form of a parabola is given by:
\[ f(x) = a(x - h)^2 + k \]
where \((h, k)\) is the vertex of the parabola and \(a\) is a coefficient that determines the width and direction of the parabola.
Given:
- Vertex: \((h, k) = (2, -1)\)
- Point on the parabola: \((4, 3)\)
Substituting the vertex into the vertex form gives us:
\[ f(x) = a(x - 2)^2 - 1 \]
Next, we need to find the value of \(a\). We can do this by substituting the point \((4, 3)\) into the equation:
Substituting \(x = 4\) and \(f(x) = 3\):
\[ 3 = a(4 - 2)^2 - 1 \]
Simplifying:
\[ 3 = a(2)^2 - 1 \] \[ 3 = 4a - 1 \]
Now, add 1 to both sides:
\[ 4 = 4a \]
Now, divide both sides by 4:
\[ a = 1 \]
Now that we have the value of \(a\), we can substitute it back into the vertex form of the equation:
\[ f(x) = 1(x - 2)^2 - 1 \]
This simplifies to:
\[ f(x) = (x - 2)^2 - 1 \]
Therefore, the equation of the parabola in vertex form is:
\[ f(x) = (x - 2)^2 - 1 \]
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