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...
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Question
Graph, find the parabola and whether it opens upwards or downwards, find the vertex, find the axis symmetry, find the x-intercepts and y-intercepts(show your work).
![The equation displayed is a quadratic function:
\[ y = -2 - 6x - x^2 \]
This equation is in the standard form of a quadratic function, which can be written as \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. In this equation:
- \( a = -1 \) (the coefficient of \( x^2 \))
- \( b = -6 \) (the coefficient of \( x \))
- \( c = -2 \) (the constant term)
### Characteristics of the Quadratic Function:
1. **Direction of the Parabola**: Since the coefficient of \( x^2 \) is negative (\( a = -1 \)), the parabola opens downwards.
2. **Vertex**: The vertex of the parabola can be found using the formula for the x-coordinate of the vertex, \( x = -\frac{b}{2a} \). Substituting \( a = -1 \) and \( b = -6 \):
\[
x = -\frac{-6}{2(-1)} = 3
\]
Substituting \( x = 3 \) back into the equation to find the y-coordinate of the vertex:
\[
y = -2 - 6(3) - (3)^2
\]
\[
y = -2 - 18 - 9
\]
\[
y = -29
\]
Therefore, the vertex is at the point \( (3, -29) \).
3. **Axis of Symmetry**: The axis of symmetry is the vertical line that passes through the vertex, \( x = 3 \).
4. **Y-intercept**: The y-intercept occurs when \( x = 0 \). Substituting \( x = 0 \) into the equation:
\[
y = -2 - 6(0) - (0)^2 = -2
\]
Thus, the y-intercept is at \( (0, -2) \).
This quadratic function represents a downward-opening parabola with specific properties determined by its coefficients.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbcea6d79-c0c2-49cd-aa31-c3eca199518f%2F6c438247-d72f-4f17-bcaa-7af8437af1ff%2F06jciye_processed.png&w=3840&q=75)
Transcribed Image Text:The equation displayed is a quadratic function:
\[ y = -2 - 6x - x^2 \]
This equation is in the standard form of a quadratic function, which can be written as \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. In this equation:
- \( a = -1 \) (the coefficient of \( x^2 \))
- \( b = -6 \) (the coefficient of \( x \))
- \( c = -2 \) (the constant term)
### Characteristics of the Quadratic Function:
1. **Direction of the Parabola**: Since the coefficient of \( x^2 \) is negative (\( a = -1 \)), the parabola opens downwards.
2. **Vertex**: The vertex of the parabola can be found using the formula for the x-coordinate of the vertex, \( x = -\frac{b}{2a} \). Substituting \( a = -1 \) and \( b = -6 \):
\[
x = -\frac{-6}{2(-1)} = 3
\]
Substituting \( x = 3 \) back into the equation to find the y-coordinate of the vertex:
\[
y = -2 - 6(3) - (3)^2
\]
\[
y = -2 - 18 - 9
\]
\[
y = -29
\]
Therefore, the vertex is at the point \( (3, -29) \).
3. **Axis of Symmetry**: The axis of symmetry is the vertical line that passes through the vertex, \( x = 3 \).
4. **Y-intercept**: The y-intercept occurs when \( x = 0 \). Substituting \( x = 0 \) into the equation:
\[
y = -2 - 6(0) - (0)^2 = -2
\]
Thus, the y-intercept is at \( (0, -2) \).
This quadratic function represents a downward-opening parabola with specific properties determined by its coefficients.
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