The function y = 2(x + 7)^2 - 4 has its vertex at the point (______, ______).
Understand the Problem
The question requires us to find the vertex of the quadratic function given in vertex form. The vertex form of a quadratic function is given by the equation y = a(x - h)^2 + k, where (h, k) represents the vertex. Here, we will identify the values of h and k from the function provided to determine the vertex.
Answer
The vertex of the quadratic function is $(3, 5)$.
Answer for screen readers
The vertex of the quadratic function is $(h, k) = (3, 5)$.
Steps to Solve
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Identify the vertex form The vertex form of a quadratic equation is given by: $$ y = a(x - h)^2 + k $$ Here, $(h, k)$ represents the vertex of the function.
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Extract values of h and k From the given quadratic function in vertex form, identify the values of $h$ and $k$. For example, if the function is represented as $$ y = 2(x - 3)^2 + 5 $$ Here, $h = 3$ and $k = 5$.
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Write down the vertex The vertex of the quadratic function can now be directly obtained from the values of $h$ and $k$. Therefore, the vertex would be: $$ (h, k) = (3, 5) $$
The vertex of the quadratic function is $(h, k) = (3, 5)$.
More Information
The vertex represents the highest or lowest point of the parabola, depending on the value of $a$. If $a > 0$, the parabola opens upwards and the vertex is the minimum point. If $a < 0$, it opens downwards and the vertex is the maximum point.
Tips
- Mixing up the signs of h: Remember that the vertex form uses $(x - h)$; if you see $(x + h)$, you need to find $h$ by taking the negative of the constant.
- Not recognizing the vertex directly from the form: The values of k and h are crucial, and they must be extracted carefully.
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