Mastering Quadratic Functions: How to Convert Standard Form to Vertex Form

Vertex form of a quadratic function

y=a(x-h)^2+k

The vertex form of a quadratic function is given by:

f(x) = a(x – h)^2 + k

where a, h, and k are constants, and (h, k) represents the vertex of the parabola.

The value of a determines the shape and direction of the parabola. If a is positive, the parabola opens upward, while if a is negative, the parabola opens downward.

The value of h represents the horizontal shift of the parabola. If h is positive, the parabola shifts to the right, while if h is negative, the parabola shifts to the left.

The value of k represents the vertical shift of the parabola. If k is positive, the parabola shifts upward, while if k is negative, the parabola shifts downward.

To convert a quadratic function from standard form to vertex form, we can complete the square by adding and subtracting (b/2a)^2 to the expression:

f(x) = ax^2 + bx + c

This gives us:

f(x) = a(x^2 + (b/2a)x) + c
f(x) = a(x^2 + (b/2a)x + (b/2a)^2 – (b/2a)^2) + c
f(x) = a(x + b/2a)^2 – a(b/2a)^2 + c
f(x) = a(x + b/2a)^2 – (b^2/4a) + c

which is in vertex form.

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