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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![## Analyzing Quadratic Graphs
### Given Graph:
1. **Graph Description:**
- The graph provided represents a parabola that opens downwards.
- The vertex is located at the point (1, 4) on the Cartesian plane.
- The parabola intersects the x-axis around (-1, 0) and (3, 0).
### Questions and Analysis:
**(a) Sign of the leading coefficient**
To determine the sign of the leading coefficient in a quadratic equation of the form \( f(x) = ax^2 + bx + c \), observe the direction of the parabola:
- Since the parabola opens downwards, the leading coefficient \(a\) is **negative**.
**(b) Vertex**
- The vertex of the parabola is the highest point, which can be identified as the point where the parabola changes direction.
- For the given graph, the vertex is at **(1, 4)**.
**(c) Axis of Symmetry**
- The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror images.
- For the given graph, the axis of symmetry is the line **x = 1**.
**(d) Intervals where \(f\) is increasing and where \(f\) is decreasing**
- The function \(f(x)\) increases as \(x\) approaches the vertex from the left.
- Interval of increase: \((-\infty, 1)\)
- The function \(f(x)\) decreases after the vertex as \(x\) moves to the right.
- Interval of decrease: \((1, \infty)\)
**(e) Domain and Range**
- **Domain**: The set of all possible \(x\)-values. Since a quadratic function is defined for all real numbers:
- Domain: All real numbers, \((-\infty, \infty)\)
- **Range**: The set of all possible \(y\)-values. For a parabola that opens downwards:
- Range: \((-\infty, 4]\)
### Example Question:
**(a) What is the sign of the leading coefficient?**
- Positive
- **Negative** (correct answer)
### Graph Details:
- **X-Axis Range**: From -5 to](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff7bd99ef-7092-4177-9c19-951c55d49391%2Fc55d36bc-74e3-4062-809a-475ca605bdfe%2Fj0ol3ho_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Analyzing Quadratic Graphs
### Given Graph:
1. **Graph Description:**
- The graph provided represents a parabola that opens downwards.
- The vertex is located at the point (1, 4) on the Cartesian plane.
- The parabola intersects the x-axis around (-1, 0) and (3, 0).
### Questions and Analysis:
**(a) Sign of the leading coefficient**
To determine the sign of the leading coefficient in a quadratic equation of the form \( f(x) = ax^2 + bx + c \), observe the direction of the parabola:
- Since the parabola opens downwards, the leading coefficient \(a\) is **negative**.
**(b) Vertex**
- The vertex of the parabola is the highest point, which can be identified as the point where the parabola changes direction.
- For the given graph, the vertex is at **(1, 4)**.
**(c) Axis of Symmetry**
- The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror images.
- For the given graph, the axis of symmetry is the line **x = 1**.
**(d) Intervals where \(f\) is increasing and where \(f\) is decreasing**
- The function \(f(x)\) increases as \(x\) approaches the vertex from the left.
- Interval of increase: \((-\infty, 1)\)
- The function \(f(x)\) decreases after the vertex as \(x\) moves to the right.
- Interval of decrease: \((1, \infty)\)
**(e) Domain and Range**
- **Domain**: The set of all possible \(x\)-values. Since a quadratic function is defined for all real numbers:
- Domain: All real numbers, \((-\infty, \infty)\)
- **Range**: The set of all possible \(y\)-values. For a parabola that opens downwards:
- Range: \((-\infty, 4]\)
### Example Question:
**(a) What is the sign of the leading coefficient?**
- Positive
- **Negative** (correct answer)
### Graph Details:
- **X-Axis Range**: From -5 to
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