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.)
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Solution

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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
  1. Identify the coefficients \( a \), \( b \), and \( c \) from the quadratic function.
  2. Calculate the x-coordinate of the vertex using \( x = -\frac{b}{2a} \).
  3. Substitute the x-coordinate back into the function to find the y-coordinate.
  4. 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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