Questions: Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=4 x^2+16 x+7 The vertex is . (Type an ordered pair.)

Find the coordinates of the vertex for the parabola defined by the given quadratic function.
f(x)=4 x^2+16 x+7

The vertex is . (Type an ordered pair.)

Transcript text: Find the coordinates of the vertex for the parabola defined by the given quadratic function. \[ f(x)=4 x^{2}+16 x+7 \] The vertex is $\square$ . (Type an ordered pair.)

Solution

Solution Steps

To find the coordinates of the vertex for the given quadratic function $ f(x) = 4x^2 + 16x + 7 $, we can use the vertex formula for a parabola in the form $ ax^2 + bx + c $. The x-coordinate of the vertex is given by $ x = -\frac{b}{2a} $. Once we have the x-coordinate, we can substitute it back into the function to find the y-coordinate.

Solution Approach

Identify the coefficients $ a $, $ b $, and $ c $ from the quadratic function.
Calculate the x-coordinate of the vertex using $ x = -\frac{b}{2a} $.
Substitute the x-coordinate back into the function to find the y-coordinate.
The vertex is the ordered pair $(x, y)$.

Step 1: Identify Coefficients

For the quadratic function $ f(x) = 4x^2 + 16x + 7 $, the coefficients are:

$ a = 4 $
$ b = 16 $
$ c = 7 $

Step 2: Calculate the x-coordinate of the Vertex

Using the formula for the x-coordinate of the vertex: \[ x = -\frac{b}{2a} = -\frac{16}{2 \cdot 4} = -2.0 \]

Step 3: Calculate the y-coordinate of the Vertex

Substituting $ x = -2.0 $ back into the function to find the y-coordinate: \[ y = f(-2.0) = 4(-2.0)^2 + 16(-2.0) + 7 = 4(4) - 32 + 7 = 16 - 32 + 7 = -9.0 \]

Final Answer

The coordinates of the vertex are $\boxed{(-2.0, -9.0)}$.

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