Determine the open intervals on which the function is increasing, decreasing, or constant. (Enter your answers using interval notation. If an answer does not exist enter DNE.) f(x) = x² - 4x increasing decreasing constant -1 y 1 2 (2.-4) 3 4 5 X 64°F Mostly cloud
Determine the open intervals on which the function is increasing, decreasing, or constant. (Enter your answers using interval notation. If an answer does not exist enter DNE.) f(x) = x² - 4x increasing decreasing constant -1 y 1 2 (2.-4) 3 4 5 X 64°F Mostly cloud
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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![### Understanding Function Intervals: Increasing, Decreasing, and Constant
**Problem Statement:**
Determine the open intervals on which the function is increasing, decreasing, or constant. (Enter your answers using interval notation. If an answer does not exist, enter DNE.)
Given function: \( f(x) = x^2 - 4x \)
**Intervals:**
- Increasing: [Input field]
- Decreasing: [Input field]
- Constant: [Input field]
**Graph Description:**
The graph of the function \( f(x) = x^2 - 4x \) is a parabola that opens upwards. The vertex of the parabola is at the point \((2, -4)\).
- The x-axis ranges from -2 to 6.
- The y-axis ranges from -8 to 6.
- The vertex at \((2, -4)\) indicates the minimum point of the function.
### Analysis of the Graph:
1. **Decreasing Interval:**
- The function \( f(x) \) is decreasing on the interval where \( x \) is less than 2. This can be observed as the slope of the graph is negative (moving downwards) as it approaches the vertex from the left (from \( -\infty \) to 2).
2. **Increasing Interval:**
- The function \( f(x) \) is increasing on the interval where \( x \) is greater than 2. This is observed as the slope of the graph turns positive (moving upwards) as it leaves the vertex moving to the right (from 2 to \( \infty \)).
3. **Constant Interval:**
- There are no intervals where the function \( f(x) \) is constant. The function is either increasing or decreasing.
### Final Intervals:
- **Increasing:** \( (2, \infty) \)
- **Decreasing:** \( (-\infty, 2) \)
- **Constant:** DNE
These intervals are found by analyzing the vertex and the direction of the parabola.
For further understanding, remember that the vertex form of a quadratic function \( ax^2 + bx + c \) can help you identify these intervals quickly:
- The vertex \((h, k)\) can be found using \( h = -\frac{b}{2a} \).
- The function is increasing on \( (h, \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F86020f22-4efe-437b-8b05-308770976ff7%2F9fb410b7-c168-4875-925a-6641489d61d5%2Fiym61gm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Function Intervals: Increasing, Decreasing, and Constant
**Problem Statement:**
Determine the open intervals on which the function is increasing, decreasing, or constant. (Enter your answers using interval notation. If an answer does not exist, enter DNE.)
Given function: \( f(x) = x^2 - 4x \)
**Intervals:**
- Increasing: [Input field]
- Decreasing: [Input field]
- Constant: [Input field]
**Graph Description:**
The graph of the function \( f(x) = x^2 - 4x \) is a parabola that opens upwards. The vertex of the parabola is at the point \((2, -4)\).
- The x-axis ranges from -2 to 6.
- The y-axis ranges from -8 to 6.
- The vertex at \((2, -4)\) indicates the minimum point of the function.
### Analysis of the Graph:
1. **Decreasing Interval:**
- The function \( f(x) \) is decreasing on the interval where \( x \) is less than 2. This can be observed as the slope of the graph is negative (moving downwards) as it approaches the vertex from the left (from \( -\infty \) to 2).
2. **Increasing Interval:**
- The function \( f(x) \) is increasing on the interval where \( x \) is greater than 2. This is observed as the slope of the graph turns positive (moving upwards) as it leaves the vertex moving to the right (from 2 to \( \infty \)).
3. **Constant Interval:**
- There are no intervals where the function \( f(x) \) is constant. The function is either increasing or decreasing.
### Final Intervals:
- **Increasing:** \( (2, \infty) \)
- **Decreasing:** \( (-\infty, 2) \)
- **Constant:** DNE
These intervals are found by analyzing the vertex and the direction of the parabola.
For further understanding, remember that the vertex form of a quadratic function \( ax^2 + bx + c \) can help you identify these intervals quickly:
- The vertex \((h, k)\) can be found using \( h = -\frac{b}{2a} \).
- The function is increasing on \( (h, \

Transcribed Image Text:This image shows a graph of a quadratic function, specifically a parabola. The parabola opens upwards, indicating that the coefficient of the squared term \( x^2 \) is positive.
### Details:
- **Axes:**
- The horizontal axis is labeled as \( x \).
- The vertical axis is labeled as \( y \).
- **Points of Intersection:**
- The parabola intersects the \( y\)-axis between -2 and -4.
- The parabola intersects the \( x\)-axis near 5 and -1.
- **Vertex:**
- The vertex of the parabola, the lowest point on the graph, is labeled as \( (2, -4) \).
The graph clearly depicts a quadratic equation in the form \( y = ax^2 + bx + c \). The specific values of \( a \), \( b \), and \( c \) are not provided in the image. The vertex form of this quadratic equation can be expressed as \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex of the parabola. For this graph, the vertex is \( (2, -4) \), leading to the vertex form \( y = a(x - 2)^2 - 4 \).
Understanding how to read such graphs is crucial for analyzing quadratic functions, which are commonly encountered in various fields of mathematics and science.
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