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The vertex form of a quadratic function is incredibly useful because it gives us direct information about the parabola's vertex, which is a key feature in the graph of the function. This form of a quadratic function is expressed as \[ y = a(x-h)^2 + k \] where \( (h, k) \) represents the vertex of the parabola. The value \( a \) can affect the width and the direction (upwards or downwards) of the opening of the parabola. If \( a \) is positive, the parabola opens upwards, and if it is negative, it opens downwards.

• \( h \) shifts the parabola left or right.
• \( k \) shifts the parabola up or down.

By rewriting a quadratic from standard form to vertex form, you gain insights into the graph's shape and orientation, simplifying both graphing and analysis tasks.

Completing the square is a method used to transform a quadratic equation into vertex form. This process involves creating a perfect square trinomial from the quadratic and linear terms of the equation. Here's how it's done:1. Take the quadratic and linear terms and place them in parentheses: \[ h(x) = (x^2 + 2x) - 48 \]2. Find the number that completes the square by taking half of the linear term's coefficient and squaring it. For \( x^2 + 2x \), you divide 2 by 2 to get 1, then square it, resulting in 1.3. Add this number inside the parentheses and subtract it immediately outside to keep the equation balanced: \[ h(x) = (x^2 + 2x + 1) - 48 - 1 \] This allows you to rewrite it as: \[ h(x) = (x + 1)^2 - 49 \]By completing the square, you can easily convert the equation to vertex form, making it simpler to find the parabola's vertex and graph the function.

The vertex of a parabola is a key point that represents either the highest or lowest point on the graph, depending on the parabola's direction. From the vertex form of a quadratic function, identifying the vertex becomes straightforward.For the function \( h(x) = (x+1)^2 - 49 \), the vertex form reveals that \[ y = a(x-h)^2 + k \], with \( h \) and \( k \) representing the coordinates of the vertex. Here, \( h = -1 \) and \( k = -49 \), so the vertex is \( (-1, -49) \).

• This vertex represents the lowest point if the parabola opens upwards and the highest if it opens downwards.
• The vertex provides insight into the parabola's position on the coordinate plane and its symmetry.

Understanding the vertex is crucial in graphing this function and analyzing its behavior. The vertex not only marks a critical point on the graph but also helps in determining the axis of symmetry of the parabola.