The Function $f(x) = -0.3(x-5)^2 + 5$ Is Graphed. What Are Some Of Its Key Features? Check All That Apply.- The Axis Of Symmetry Is $x = 5$.- The Domain Is $\{x \mid X \text{ Is A Real Number}\}$.- The Function Is Increasing

The Function $f(x) = -0.3(x-5)^2 + 5$: Unveiling its Key Features

The given function $f(x) = -0.3(x-5)^2 + 5$ is a quadratic function in the form of $f(x) = a(x-h)^2 + k$, where $a$, $h$, and $k$ are constants. This function is graphed, and we are tasked with identifying some of its key features. In this article, we will delve into the properties of this function and determine which of the given options are correct.

The axis of symmetry is a line that passes through the vertex of the parabola and is perpendicular to the axis of the parabola. In the case of the function $f(x) = -0.3(x-5)^2 + 5$, the vertex is at the point $(5, 5)$. Since the vertex is at $x = 5$, the axis of symmetry is also at $x = 5$. Therefore, the correct answer is:

• The axis of symmetry is $x = 5$.

The domain of a function is the set of all possible input values for which the function is defined. In the case of the function $f(x) = -0.3(x-5)^2 + 5$, the function is defined for all real numbers. Therefore, the domain of the function is $\{x \mid x \text{ is a real number}\}$. The correct answer is:

• The domain is $\{x \mid x \text{ is a real number}\}$.

An increasing function is a function that increases as the input value increases. In the case of the function $f(x) = -0.3(x-5)^2 + 5$, the coefficient of the squared term is negative, which means that the function is concave down. This means that the function decreases as the input value increases. Therefore, the correct answer is:

• The function is decreasing.

In conclusion, the key features of the function $f(x) = -0.3(x-5)^2 + 5$ are:

• The axis of symmetry is $x = 5$.
• The domain is $\{x \mid x \text{ is a real number}\}$.
• The function is decreasing.

These features are a result of the function's quadratic form and the values of its coefficients. Understanding these features is essential for analyzing and graphing the function.

To graph the function $f(x) = -0.3(x-5)^2 + 5$, we can use the following steps:

1. Identify the vertex: The vertex of the parabola is at the point $(5, 5)$.
2. Identify the axis of symmetry: The axis of symmetry is at $x = 5$.
3. Identify the direction of the parabola: The parabola opens downward since the coefficient of the squared term is negative.
4. Plot the parabola: Plot the parabola using the vertex and the axis of symmetry as reference points.

Let's consider an example to illustrate the graphing process. Suppose we want to graph the function $f(x) = -0.3(x-5)^2 + 5$.

1. Identify the vertex: The vertex of the parabola is at the point $(5, 5)$.
2. Identify the axis of symmetry: The axis of symmetry is at $x = 5$.
3. Identify the direction of the parabola: The parabola opens downward since the coefficient of the squared term is negative.
4. Plot the parabola: Plot the parabola using the vertex and the axis of symmetry as reference points.

The resulting graph will be a downward-opening parabola with its vertex at the point $(5, 5)$ and its axis of symmetry at $x = 5$.

In conclusion, the function $f(x) = -0.3(x-5)^2 + 5$ has the following key features:

• The axis of symmetry is $x = 5$.
• The domain is $\{x \mid x \text{ is a real number}\}$.
• The function is decreasing.

These features are a result of the function's quadratic form and the values of its coefficients. Understanding these features is essential for analyzing and graphing the function.
The Function $f(x) = -0.3(x-5)^2 + 5$: A Q&A Guide

In our previous article, we explored the key features of the function $f(x) = -0.3(x-5)^2 + 5$. In this article, we will answer some frequently asked questions about this function to provide a deeper understanding of its properties and behavior.

Q: What is the vertex of the parabola?

A: The vertex of the parabola is at the point $(5, 5)$.

Q: What is the axis of symmetry?

A: The axis of symmetry is at $x = 5$.

Q: Is the function increasing or decreasing?

A: The function is decreasing.

Q: What is the domain of the function?

A: The domain of the function is $\{x \mid x \text{ is a real number}\}$.

Q: How do I graph the function?

A: To graph the function, follow these steps:

1. Identify the vertex: The vertex of the parabola is at the point $(5, 5)$.
2. Identify the axis of symmetry: The axis of symmetry is at $x = 5$.
3. Identify the direction of the parabola: The parabola opens downward since the coefficient of the squared term is negative.
4. Plot the parabola: Plot the parabola using the vertex and the axis of symmetry as reference points.

Q: What is the significance of the coefficient $-0.3$?

A: The coefficient $-0.3$ determines the direction and steepness of the parabola. Since the coefficient is negative, the parabola opens downward. The magnitude of the coefficient affects the steepness of the parabola.

Q: Can I use this function in real-world applications?

A: Yes, this function can be used in various real-world applications, such as modeling population growth, predicting stock prices, or analyzing data in science and engineering.

Q: How do I find the x-intercepts of the parabola?

A: To find the x-intercepts, set the function equal to zero and solve for $x$. In this case, we have:

$$-0.3(x-5)^2 + 5 = 0$$

Solving for $x$, we get:

$$x = 5 \pm \sqrt{\frac{15}{0.3}}$$

The x-intercepts are at $x = 5 \pm \sqrt{50}$.

Q: Can I use this function to model a physical system?

A: Yes, this function can be used to model a physical system, such as a spring-mass system or a pendulum. The function can be used to describe the motion of the system and predict its behavior.

In conclusion, the function $f(x) = -0.3(x-5)^2 + 5$ has several key features, including a vertex at $(5, 5)$, an axis of symmetry at $x = 5$, and a decreasing behavior. Understanding these features is essential for analyzing and graphing the function. We hope this Q&A guide has provided a deeper understanding of the function and its properties.