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.)
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 \(\boxed{(-2.0, -9.0)}\).