To convert the quadratic function f(x) = x² + 8x + 3 into vertex form, we need to complete the square. Vertex form is typically expressed as:
f(x) = a(x – h)² + k
where (h, k) is the vertex of the parabola represented by the quadratic function.
Here are the steps to achieve this:
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Start with the original function:
f(x) = x² + 8x + 3
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Focus on the x² + 8x part. To complete the square, we need to find a number that makes this expression a perfect square trinomial. Divide the coefficient of x (which is 8) by 2 and square it:
(8 / 2)² = 4² = 16
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Add and subtract this value inside the function:
f(x) = x² + 8x + 16 – 16 + 3
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Now rewrite the equation, grouping the perfect square trinomial:
f(x) = (x + 4)² – 13
We added 16 and subtracted 16 to keep the equation balanced.
At this point, we have the quadratic function in vertex form:
f(x) = (x + 4)² – 13
From this form, we can see that the vertex of the parabola is at the point (-4, -13). This means the graph of the function opens upwards with its lowest point (the vertex) located at (-4, -13).
In summary, the vertex form of the quadratic function f(x) = x² + 8x + 3 is:
f(x) = (x + 4)² – 13