Intervals on which the Function Increases and Decreases

Oct 13, 2025 by miodeya.com 53 views

Introduction

In this article, we will explore the graph of the function $f(x)=-2|x+1|+2$ and analyze its key features. The function is a quadratic function in the form of $f(x)=a|x-h|+k$ , where $a$ , $h$ , and $k$ are constants. We will graph the function and observe its vertex location, vertex (maximum or minimum), intervals on which the function increases and decreases, end behavior, and intercepts.

Graphing the Function

To graph the function $f(x)=-2|x+1|+2$ , we can start by identifying the vertex of the function. The vertex of the function is located at the point $(h, k)$ , where $h$ and $k$ are the constants in the function. In this case, the vertex is located at the point $(-1, 2)$ .

import numpy as np
import matplotlib.pyplot as plt
def f(x):
return -2 * np.abs(x + 1) + 2
x = np.linspace(-5, 5, 400)
y = f(x)
plt.plot(x, y)
plt.title('Graph of  $f(x)=-2|x+1|+2$ ')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.grid(True)
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.show()

Vertex Location

The vertex of the function $f(x)=-2|x+1|+2$ is located at the point $(-1, 2)$ . This means that the function has a maximum value of $2$ at $x=-1$ .

Vertex (Maximum or Minimum)

Since the function has a maximum value of $2$ at $x=-1$ , the vertex is a maximum point.

Intervals on which the Function Increases and Decreases

To determine the intervals on which the function increases and decreases, we can use the first derivative of the function. The first derivative of the function is given by:

f'(x)=-2\text{sign}(x+1)

where $\text{sign}(x+1)$ is the sign function, which is equal to $1$ if $x+1>0$ and $-1$ if $x+1<0$ .

import numpy as np
def f_prime(x):
return -2 * np.sign(x + 1)
x = np.linspace(-5, 5, 400)
y = f_prime(x)
plt.plot(x, y)
plt.title('First Derivative of  $f(x)=-2|x+1|+2$ ')
plt.xlabel('x')
plt.ylabel('f&#39;(x)')
plt.grid(True)
plt.axhline(0, color='black')
plt.axvline(0, color='black')
plt.show()

From the first derivative, we can see that the function increases on the interval $(-\infty, -1)$ and decreases on the interval $(-1, \infty)$ .

End Behavior

To determine the end behavior of the function, we can use the fact that the function is a quadratic function. The end behavior of a quadratic function is determined by the sign of the coefficient of the squared term. In this case, the coefficient of the squared term is $-2$ , which is negative. Therefore, the function has a downward opening shape.

Intercepts

To find the x-intercepts of the function, we can set the function equal to zero and solve for $x$ .

-2|x+1|+2=0

Solving for $x$ , we get:

|x+1|=1

This gives us two possible solutions:

x+1=1 \Rightarrow x=0

x+1=-1 \Rightarrow x=-2

Therefore, the x-intercepts of the function are $x=0$ and $x=-2$ .

Conclusion

Q: What is the vertex location of the function $f(x)=-2|x+1|+2$ ?

A: The vertex location of the function $f(x)=-2|x+1|+2$ is at the point $(-1, 2)$ .

Q: What is the vertex (maximum or minimum) of the function $f(x)=-2|x+1|+2$ ?

A: The vertex of the function $f(x)=-2|x+1|+2$ is a maximum point, with a maximum value of $2$ at $x=-1$ .

Q: On which intervals does the function $f(x)=-2|x+1|+2$ increase and decrease?

A: The function $f(x)=-2|x+1|+2$ increases on the interval $(-\infty, -1)$ and decreases on the interval $(-1, \infty)$ .

Q: What is the end behavior of the function $f(x)=-2|x+1|+2$ ?

A: The function $f(x)=-2|x+1|+2$ has a downward opening shape, meaning that it decreases as $x$ increases.

Q: What are the x-intercepts of the function $f(x)=-2|x+1|+2$ ?

A: The x-intercepts of the function $f(x)=-2|x+1|+2$ are at $x=0$ and $x=-2$ .

Q: How can I graph the function $f(x)=-2|x+1|+2$ ?

A: You can graph the function $f(x)=-2|x+1|+2$ using a graphing calculator or a computer algebra system. Alternatively, you can use the code provided earlier to generate a plot of the function.

Q: What is the significance of the absolute value function in the graph of $f(x)=-2|x+1|+2$ ?

A: The absolute value function in the graph of $f(x)=-2|x+1|+2$ creates a V-shaped graph with a maximum point at $x=-1$ . The absolute value function is responsible for the sharp turn in the graph at $x=-1$ .

Q: Can I use the graph of $f(x)=-2|x+1|+2$ to determine the value of the function at a given point?

A: Yes, you can use the graph of $f(x)=-2|x+1|+2$ to determine the value of the function at a given point. Simply locate the point on the graph corresponding to the given value of $x$ and read off the value of the function at that point.

Q: How can I use the graph of $f(x)=-2|x+1|+2$ to solve equations involving the function?

A: You can use the graph of $f(x)=-2|x+1|+2$ to solve equations involving the function by locating the points on the graph that satisfy the equation. For example, to solve the equation $f(x)=-2|x+1|+2=0$ , you can locate the points on the graph where the function is equal to zero.

Q: Can I use the graph of $f(x)=-2|x+1|+2$ to determine the domain and range of the function?

A: Yes, you can use the graph of $f(x)=-2|x+1|+2$ to determine the domain and range of the function. The domain of the function is the set of all values of $x$ for which the function is defined, while the range of the function is the set of all possible values of the function. From the graph, you can see that the domain of the function is all real numbers, and the range of the function is all real numbers greater than or equal to $-2$ .