Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section: Chapter Questions
Problem 7DE
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![### Finding the Vertex of a Parabola
In this exercise, you are asked to find the coordinates of the vertex for a parabola defined by the given quadratic function.
#### Question:
1. \( f(x) = -x^2 - 4x - 9 \)
Below the quadratic function, a space is provided for writing the answer.
To find the vertex of a parabola given by a quadratic function \( f(x) = ax^2 + bx + c \):
- The x-coordinate of the vertex can be found using the formula \( x = -\frac{b}{2a} \).
- Once you have the x-coordinate, substitute it back into the original function to find the y-coordinate.
In this example, the quadratic function \( f(x) = -x^2 - 4x - 9 \) is given:
- Here, \( a = -1 \), \( b = -4 \), and \( c = -9 \).
First, find the x-coordinate of the vertex:
\[ x = -\frac{-4}{2(-1)} = \frac{4}{-2} = -2 \]
Next, substitute \( x = -2 \) back into the function to find the y-coordinate:
\[ f(-2) = -(-2)^2 - 4(-2) - 9 = -4 + 8 - 9 = -5 \]
Thus, the coordinates of the vertex are \( (-2, -5) \).
You can now fill in the provided space with the answer:
\[ 1) \quad (-2, -5) \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77a13500-3283-4aa2-b558-e9fbf21d692d%2F6ebf7555-eda4-4285-bf2d-2362ad11807c%2Fcv2iw6c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding the Vertex of a Parabola
In this exercise, you are asked to find the coordinates of the vertex for a parabola defined by the given quadratic function.
#### Question:
1. \( f(x) = -x^2 - 4x - 9 \)
Below the quadratic function, a space is provided for writing the answer.
To find the vertex of a parabola given by a quadratic function \( f(x) = ax^2 + bx + c \):
- The x-coordinate of the vertex can be found using the formula \( x = -\frac{b}{2a} \).
- Once you have the x-coordinate, substitute it back into the original function to find the y-coordinate.
In this example, the quadratic function \( f(x) = -x^2 - 4x - 9 \) is given:
- Here, \( a = -1 \), \( b = -4 \), and \( c = -9 \).
First, find the x-coordinate of the vertex:
\[ x = -\frac{-4}{2(-1)} = \frac{4}{-2} = -2 \]
Next, substitute \( x = -2 \) back into the function to find the y-coordinate:
\[ f(-2) = -(-2)^2 - 4(-2) - 9 = -4 + 8 - 9 = -5 \]
Thus, the coordinates of the vertex are \( (-2, -5) \).
You can now fill in the provided space with the answer:
\[ 1) \quad (-2, -5) \]
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