Which graph represents the solution of y >= x ^ 2 + 2 ? **Which graph represents the solution of y ≥ x² + 2?**This image shows two graphs, both illustrating a parabola and shaded regions, with the aim to determine which graph accurately represents the solution for the inequality \( y \ge x^2 + 2 \).### First Graph:- **Parabola:** The parabola represented by the equation \( y = x^2 + 2 \) is depicted with a dashed red line.- **Shaded Region:** The region above the parabola (where \( y \ge x^2 + 2 \)) is shaded in a solid brown color.- **Axes:** The graph has horizontal and vertical axes, with the horizontal axis (x-axis) ranging from -5 to 5, and the vertical axis (y-axis) ranging from 0 to 9.### Second Graph:- **Parabola:** Similar to the first graph, the parabola is represented by the equation \( y = x^2 + 2 \) and is outlined with a dashed red line.- **Shaded Region:** Contrary to the correct representation, the shaded region here is below the parabola, in a solid brown color.- **Axes:** This graph also features horizontal and vertical axes, with the x-axis ranging from -5 to 5, and the y-axis from 0 to 9.### Explanation:For the inequality \( y \ge x^2 + 2 \), the solution includes all points on and above the parabola \( y = x^2 + 2 \). Thus, the correct representation is the one where the region above the curve is shaded.**Correct Graph:**The first graph accurately represents the solution for \( y \ge x^2 + 2 \), as the area above the curve (including the curve itself) is shaded. ### Understanding Graphs of Quadratic FunctionsQuadratic functions, which are polynomial functions of degree 2, can be represented graphically by parabolas. Here, we analyze the graphs of two quadratic functions shown in the images.#### First Graph:- **Axes**: The horizontal axis is labeled x, and the vertical axis is labeled y. Both axes are marked with increments.- **Graph Description**: The parabola opens upwards, signifying that the coefficient of the quadratic term is positive. The vertex of the parabola, which is the minimum point, appears to be at (0, 3). The graph is shaded in red from the vertex upwards.Key Characteristics:- **Vertex**: (0, 3)- **Direction**: Upward- **X-Intercepts**: The graph does not cross the x-axis, implying complex roots or no real roots.- **Y-Intercept**: The graph crosses the y-axis at (0, 3).This indicates that the quadratic function may have the general form of \( y = ax^2 + bx + c \) where the vertex form is \( y = a(x-h)^2 + k \) and here \( h = 0, k = 3 \).#### Second Graph:- **Axes**: Similar to the first graph, the horizontal axis is labeled x, and the vertical axis is labeled y. Both axes are marked with increments.- **Graph Description**: The parabola in this graph opens downwards, indicating that the coefficient of the quadratic term is negative. The vertex of the parabola is at (0, 3). The graph is shaded in red from the vertex downwards.Key Characteristics:- **Vertex**: (0, 3)- **Direction**: Downward- **X-Intercepts**: The graph does not cross the x-axis, implying no real roots or complex roots.- **Y-Intercept**: The graph crosses the y-axis at (0, 3).This signifies that the quadratic function is likely in the form \( y = -ax^2 + bx + c \) where the vertex form is \( y = -a(x-h)^2 + k \) and, similarly, \( h = 0, k = 3 \).### Summary:Both graphs illustrate key properties of quadratic functions through their graphical representations:1. **Vertex**: The turning point of the

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Which graph represents the solution of y >= x ^ 2 + 2 ?

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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Question

Which graph represents the solution of y >= x ^ 2 + 2 ?

**Which graph represents the solution of y ≥ x² + 2?**

This image shows two graphs, both illustrating a parabola and shaded regions, with the aim to determine which graph accurately represents the solution for the inequality \( y \ge x^2 + 2 \).

### First Graph:

- **Parabola:** The parabola represented by the equation \( y = x^2 + 2 \) is depicted with a dashed red line.
- **Shaded Region:** The region above the parabola (where \( y \ge x^2 + 2 \)) is shaded in a solid brown color.
- **Axes:** The graph has horizontal and vertical axes, with the horizontal axis (x-axis) ranging from -5 to 5, and the vertical axis (y-axis) ranging from 0 to 9.

### Second Graph:

- **Parabola:** Similar to the first graph, the parabola is represented by the equation \( y = x^2 + 2 \) and is outlined with a dashed red line.
- **Shaded Region:** Contrary to the correct representation, the shaded region here is below the parabola, in a solid brown color.
- **Axes:** This graph also features horizontal and vertical axes, with the x-axis ranging from -5 to 5, and the y-axis from 0 to 9.

### Explanation:

For the inequality \( y \ge x^2 + 2 \), the solution includes all points on and above the parabola \( y = x^2 + 2 \). Thus, the correct representation is the one where the region above the curve is shaded.

**Correct Graph:**

The first graph accurately represents the solution for \( y \ge x^2 + 2 \), as the area above the curve (including the curve itself) is shaded.

### Understanding Graphs of Quadratic Functions

Quadratic functions, which are polynomial functions of degree 2, can be represented graphically by parabolas. Here, we analyze the graphs of two quadratic functions shown in the images.

#### First Graph:
- **Axes**: The horizontal axis is labeled x, and the vertical axis is labeled y. Both axes are marked with increments.
- **Graph Description**: The parabola opens upwards, signifying that the coefficient of the quadratic term is positive. The vertex of the parabola, which is the minimum point, appears to be at (0, 3). The graph is shaded in red from the vertex upwards.

Key Characteristics:
- **Vertex**: (0, 3)
- **Direction**: Upward
- **X-Intercepts**: The graph does not cross the x-axis, implying complex roots or no real roots.
- **Y-Intercept**: The graph crosses the y-axis at (0, 3).

This indicates that the quadratic function may have the general form of \( y = ax^2 + bx + c \) where the vertex form is \( y = a(x-h)^2 + k \) and here \( h = 0, k = 3 \).

#### Second Graph:
- **Axes**: Similar to the first graph, the horizontal axis is labeled x, and the vertical axis is labeled y. Both axes are marked with increments.
- **Graph Description**: The parabola in this graph opens downwards, indicating that the coefficient of the quadratic term is negative. The vertex of the parabola is at (0, 3). The graph is shaded in red from the vertex downwards.

Key Characteristics:
- **Vertex**: (0, 3)
- **Direction**: Downward
- **X-Intercepts**: The graph does not cross the x-axis, implying no real roots or complex roots.
- **Y-Intercept**: The graph crosses the y-axis at (0, 3).

This signifies that the quadratic function is likely in the form \( y = -ax^2 + bx + c \) where the vertex form is \( y = -a(x-h)^2 + k \) and, similarly, \( h = 0, k = 3 \).

### Summary:

Both graphs illustrate key properties of quadratic functions through their graphical representations:
1. **Vertex**: The turning point of the

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