Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=−2(x+1)²+5

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Identify the form of the quadratic function. The given function is in vertex form: \(f(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex.

Rewrite the function to match the vertex form template. The function is \(f(x) = -2(x + 1)^2 + 5\), which can be seen as \(f(x) = -2(x - (-1))^2 + 5\).

From the expression, determine the vertex coordinates by comparing to \(f(x) = a(x - h)^2 + k\). Here, \(h = -1\) and \(k = 5\).

Therefore, the vertex of the parabola is at the point \((-1, 5)\).

Remember that the coefficient \(a = -2\) indicates the parabola opens downward, but this does not affect the vertex coordinates.

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주요 개념

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Vertex Form of a Quadratic Function

The vertex form of a quadratic function is expressed as f(x) = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola. This form makes it easy to identify the vertex directly from the equation without further calculations.

Effect of Parameters a, h, and k on the Parabola

In the vertex form, 'a' determines the parabola's direction and width: if a is negative, the parabola opens downward; if positive, upward. The values 'h' and 'k' shift the parabola horizontally and vertically, respectively, locating the vertex at (h, k).

Finding the Vertex Coordinates

To find the vertex coordinates from a quadratic in vertex form, identify h and k from the expression f(x) = a(x - h)^2 + k. Note the sign inside the parentheses is opposite to the vertex's x-coordinate. The vertex is then (h, k), which gives the maximum or minimum point of the parabola.

Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=−2(x+1)2+5