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Convert f(x) into vertex form, then identify the vertex.

f(x) = -5x^(2) -40x-92

User Kendell
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1 Answer

4 votes

Answer:

Vertex form:
\sf f(x) = -5(x + 4)^2 - 12

Vertex: (-4, -12)

Explanation:

To convert the quadratic function
f(x) = -5x^2 - 40x - 92 into vertex form
\sf f(x) = a(x - h)^2 + k, we need to complete the square.

The vertex form of a quadratic equation allows us to easily identify the vertex.

Factor out the common coefficient (-5) from the x² and x terms:


\sf f(x) = -5(x^2 + 8x) - 92

To complete the square, we need to add and subtract a constant inside the parentheses such that it will be a perfect square trinomial.

To do this, take half of the coefficient of the x term
\sf \left((8)/(2) = 4\right), square it
\sf (4^2 = 16) , and add it inside the parentheses:


\sf f(x) = -5(x^2 + 8x + 16 - 16) - 92

Rewrite the expression and simplify:


\sf f(x) = -5((x^2 + 8x + 16) - 16) - 92

Now, we have a perfect square trinomial inside the parentheses:


\sf f(x) = -5((x + 4)^2 - 16) - 92

Distribute the -5 to both terms inside the parentheses:


\sf f(x) = -5(x + 4)^2 + 80 - 92

Simplify further:


\sf f(x) = -5(x + 4)^2 - 12

Now, the function is in vertex form
\sf f(x) = a(x - h)^2 + k, where the vertex is at the point (h, k).

In this case, the vertex is (-4, -12).

So, the vertex form is
\sf f(x) = -5(x + 4)^2 - 12, and the vertex is (-4, -12).

User Nick Radford
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7.3k points

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