Questions: Find the coordinates of the vertex, by hand, of the graph of the function. Tell whether the vertex is a maximum or a minimum. y=2x^2+4x+7

Find the coordinates of the vertex, by hand, of the graph of the function. Tell whether the vertex is a maximum or a minimum. y=2x^2+4x+7
Transcript text: Find the coordinates of the vertex, by hand, of the graph of the function. Tell whether the vertex is a maximum or a minimum. \[ y=2 x^{2}+4 x+7 \]
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Solution

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Solution Steps

To find the vertex of a quadratic function in the form \( y = ax^2 + bx + c \), we use the vertex formula \( x = -\frac{b}{2a} \). Once we have the x-coordinate, we substitute it back into the function to find the y-coordinate. The vertex will be a minimum if \( a > 0 \) and a maximum if \( a < 0 \).

Step 1: Find the x-coordinate of the Vertex

To find the x-coordinate of the vertex for the quadratic function \( y = 2x^2 + 4x + 7 \), we use the formula: \[ x = -\frac{b}{2a} \] Substituting \( a = 2 \) and \( b = 4 \): \[ x = -\frac{4}{2 \cdot 2} = -\frac{4}{4} = -1 \]

Step 2: Find the y-coordinate of the Vertex

Next, we substitute \( x = -1 \) back into the function to find the y-coordinate: \[ y = 2(-1)^2 + 4(-1) + 7 \] Calculating this gives: \[ y = 2(1) - 4 + 7 = 2 - 4 + 7 = 5 \]

Step 3: Determine if the Vertex is a Maximum or Minimum

Since the coefficient \( a = 2 \) is greater than 0, the vertex represents a minimum point.

Final Answer

The coordinates of the vertex are \( (-1, 5) \) and it is a minimum. Thus, the answer is: \[ \boxed{(-1, 5) \text{ ; minimum}} \]

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