**Unlocking the Secrets of Graphical Functions**

Have you ever encountered a puzzling graph and wondered, “Which function is hiding behind this curve?” If so, you’re not alone. Identifying the function that corresponds to a given graph is a fundamental skill in mathematics and can be applied in various fields. This blog post will guide you through the process of choosing the function whose graph is given below, empowering you to decode the hidden relationships in graphical data.

Navigating the complex world of functions can be challenging, but understanding the underlying principles can simplify the task. Just as a detective unravels clues to solve a mystery, we can analyze the key features of a graph to determine the corresponding function.

**Matching Function to Graph: A Precise Approach**

To effectively choose the function whose graph is given below, it’s essential to identify the graph’s key attributes, such as its shape, symmetry, end behavior, and any specific points. By carefully examining these characteristics, we can narrow down the possibilities and pinpoint the correct function. The process involves a combination of visual analysis and algebraic manipulations, leading us to the function that accurately represents the graphical data.

**Key to Unlocking Graphical Functions: A Simple Guide**

Equipped with the necessary knowledge and techniques, you’ll be able to confidently identify the function associated with any given graph. This will not only enhance your mathematical prowess but also broaden your problem-solving abilities in diverse applications. Remember, the journey of understanding graphical functions begins with a single graph, and with each step, you’ll become more proficient in deciphering the language of mathematics.

## Understanding the Function Represented by the Given Graph

**Introduction**

The provided graph depicts the relationship between two variables, revealing a specific mathematical function. This article aims to identify the function that best represents the graph, providing a detailed analysis of its characteristics.

**1. Linear Function**

A linear function takes the form y = mx + b, where m is the slope and b is the y-intercept. The graph of a linear function is a straight line.

**2. Quadratic Function**

A quadratic function takes the form y = ax² + bx + c, where a, b, and c are constants. The graph of a quadratic function is a parabola.

**3. Exponential Function**

An exponential function takes the form y = a^x, where a is a positive constant. The graph of an exponential function is a curve that increases or decreases rapidly.

**4. Logarithmic Function**

A logarithmic function takes the form y = logₐ(x), where a is a positive constant. The graph of a logarithmic function is a curve that decreases or increases gradually.

**Analysis of the Graph**

Examining the graph, we observe the following characteristics:

**Shape:**The graph is a smooth, continuous curve that appears parabolic.**Vertex:**The highest point of the parabola is located at (0, 2).**Intercepts:**The graph intercepts the x-axis at x = -1 and x = 1.

**Identification of the Function**

Based on these characteristics, we can eliminate the linear and logarithmic functions, as they do not exhibit a parabolic shape. The remaining options are the quadratic and exponential functions.

**Quadratic Function Justification:**

The graph’s parabolic shape and vertex strongly suggest a quadratic function. The vertex form of a quadratic function is y = a(x – h)² + k, where (h, k) is the vertex. In our case, (h, k) = (0, 2), so the function is in the form y = a(x – 0)² + 2 = ax² + 2.

**Determining the Value of “a”**

We can determine the value of “a” by using the x-intercepts. At x = -1, y = 0, so we have:

```
0 = a(-1)² + 2
0 = a + 2
a = -2
```

**Conclusion**

Thus, the function represented by the given graph is **y = -2x² + 2**. This quadratic function accurately reflects the parabolic shape, vertex, and intercepts of the graph.

**FAQs**

**What is the slope of the function?**

- The function is a parabola and therefore does not have a slope.

**What is the y-intercept of the function?**

- The y-intercept is 2, which is the value of y when x is 0.

**What is the domain of the function?**

- The domain of the function is all real numbers since the parabola extends infinitely in both directions.

**What is the range of the function?**

- The range of the function is y ≥ 2 since the parabola opens upwards and has a minimum at y = 2.

**What is the equation of the axis of symmetry for the parabola?**

- The axis of symmetry is a vertical line passing through the vertex, which is x = 0.

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