Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
![**Exercise: Finding the Equation of a Parabola**
**Problem Statement:**
II. Find an equation of the parabola graphed.
2.
**Graph Description:**
- The graph presents a parabola opening downwards.
- The vertex of the parabola is at the point (-3, 4).
- It intersects the y-axis at approximately y = -4.
- The x-intercepts are approximately at x = -5 and x = -1, confirming a downward opening.
**Steps to Find the Equation:**
1. **Vertex Form of a Parabola:**
- The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex.
- In this graph, the vertex \((h, k)\) is \((-3, 4)\).
2. **Finding 'a':**
- Using another point on the graph to solve for \( a \). For example, use the point (-1, 0), which lies on the parabola.
- Substitute \((x, y) = (-1, 0)\) into the vertex form equation:
\[
0 = a(-1 + 3)^2 + 4
\]
\[
0 = 4a + 4
\]
\[
-4 = 4a
\]
\[
a = -1
\]
3. **Equation of the Parabola:**
- Substitute \( a\), \( h\), and \( k\) into the vertex form:
\[
y = -1(x + 3)^2 + 4
\]
This is the equation of the parabola that matches the graph provided.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F92b9fa95-2e7d-446b-a380-41f972519da6%2F7e37dd15-9506-4fd9-b665-56f88551e10a%2F0gkr86g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Exercise: Finding the Equation of a Parabola**
**Problem Statement:**
II. Find an equation of the parabola graphed.
2.
**Graph Description:**
- The graph presents a parabola opening downwards.
- The vertex of the parabola is at the point (-3, 4).
- It intersects the y-axis at approximately y = -4.
- The x-intercepts are approximately at x = -5 and x = -1, confirming a downward opening.
**Steps to Find the Equation:**
1. **Vertex Form of a Parabola:**
- The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex.
- In this graph, the vertex \((h, k)\) is \((-3, 4)\).
2. **Finding 'a':**
- Using another point on the graph to solve for \( a \). For example, use the point (-1, 0), which lies on the parabola.
- Substitute \((x, y) = (-1, 0)\) into the vertex form equation:
\[
0 = a(-1 + 3)^2 + 4
\]
\[
0 = 4a + 4
\]
\[
-4 = 4a
\]
\[
a = -1
\]
3. **Equation of the Parabola:**
- Substitute \( a\), \( h\), and \( k\) into the vertex form:
\[
y = -1(x + 3)^2 + 4
\]
This is the equation of the parabola that matches the graph provided.
Expert Solution
Step 1
The equation of the parabola: - A parabola is a curve equation in which a point on the curve is equidistant from both a fixed point and a fixed-line. The fixed point is known as the parabola's focus, and the fixed-line is known as the parabola's directrix.
Parabola equation: -
Where is a vertex.
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