## 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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(a) What is the sign of the leading coefficient? O Positive O Negative

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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Polynomials

Try substituting 1. You will get 6 and not 0. Try other real values also. What About -2? Let us check.

Question

## 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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