Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Writing the Equation of a Parabola in Standard Form
To find the equation of the parabola depicted in the graph, we first need to observe the critical points provided.
**Equation Format:**
The standard form of the equation of a parabola is given by:
\[ f(x) = ax^2 + bx + c \]
**Given Points:**
From the graph, the following points on the parabola are provided:
- Vertex: \((-3, -2)\)
- A point on the graph: \((0, 7)\)
**Steps to Determine the Equation:**
1. **Vertex Substitution**:
Given that \((h, k) = (-3, -2)\), we can start with the vertex form of the parabola equation:
\[ f(x) = a(x + 3)^2 - 2 \]
2. **Using Another Point**:
Substitute point \((0, 7)\) into the equation:
\[ 7 = a(0 + 3)^2 - 2 \]
Simplify and solve for \(a\):
\[ 7 = 9a - 2 \]
\[ 9a = 9 \]
\[ a = 1 \]
3. **Complete Equation**:
Now substituting \(a = 1\) back into the vertex form, we get:
\[ f(x) = (x + 3)^2 - 2 \]
Expand to standard form:
\[ f(x) = x^2 + 6x + 9 - 2 \]
\[ f(x) = x^2 + 6x + 7 \]
So the equation in standard form is:
\[ f(x) = x^2 + 6x + 7 \]
#### Graph Explanation:
The graph displays a parabola which opens upwards. Its vertex is at \((-3, -2)\), indicating the lowest point of the curve. Another key point given is \((0, 7)\), which lies on the parabola. Both the x-axis and y-axis range from -10 to 10, providing a clear view of the parabola's shape and key points.
By deriving the equation step-by-step and explaining the graph, we offer a comprehensive understanding of how to write the equation of a parabola in standard form.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b914819-4152-4353-8ebc-b14bc0f05ba7%2F5f1637d3-dd31-4a33-8a9d-f9ccb8657b5d%2Ft3mf13a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Writing the Equation of a Parabola in Standard Form
To find the equation of the parabola depicted in the graph, we first need to observe the critical points provided.
**Equation Format:**
The standard form of the equation of a parabola is given by:
\[ f(x) = ax^2 + bx + c \]
**Given Points:**
From the graph, the following points on the parabola are provided:
- Vertex: \((-3, -2)\)
- A point on the graph: \((0, 7)\)
**Steps to Determine the Equation:**
1. **Vertex Substitution**:
Given that \((h, k) = (-3, -2)\), we can start with the vertex form of the parabola equation:
\[ f(x) = a(x + 3)^2 - 2 \]
2. **Using Another Point**:
Substitute point \((0, 7)\) into the equation:
\[ 7 = a(0 + 3)^2 - 2 \]
Simplify and solve for \(a\):
\[ 7 = 9a - 2 \]
\[ 9a = 9 \]
\[ a = 1 \]
3. **Complete Equation**:
Now substituting \(a = 1\) back into the vertex form, we get:
\[ f(x) = (x + 3)^2 - 2 \]
Expand to standard form:
\[ f(x) = x^2 + 6x + 9 - 2 \]
\[ f(x) = x^2 + 6x + 7 \]
So the equation in standard form is:
\[ f(x) = x^2 + 6x + 7 \]
#### Graph Explanation:
The graph displays a parabola which opens upwards. Its vertex is at \((-3, -2)\), indicating the lowest point of the curve. Another key point given is \((0, 7)\), which lies on the parabola. Both the x-axis and y-axis range from -10 to 10, providing a clear view of the parabola's shape and key points.
By deriving the equation step-by-step and explaining the graph, we offer a comprehensive understanding of how to write the equation of a parabola in standard form.
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