Understanding The Characteristics Of Function Graphs: Lesson 1.2 Answers
When it comes to understanding mathematical functions, one of the essential concepts to grasp is the characteristics of function graphs. These characteristics can help us identify the properties of a function, its behavior, and its values. In this article, we will delve deeper into the answers of Lesson 1.2 about the characteristics of function graphs, so you can have a better understanding of this fundamental concept in mathematics.
What are Function Graphs?
Before we dive into the characteristics of function graphs, let's first define what a function graph is. A function graph is a visual representation of a mathematical function that shows how its input values (x) relate to its output values (y). A function graph is typically plotted on a Cartesian coordinate system, where the horizontal axis represents the input values, and the vertical axis represents the output values.
What are the Characteristics of Function Graphs?
There are several characteristics of function graphs that we need to understand, including:
Domain and Range
The domain is the set of all possible input values (x) of a function, while the range is the set of all possible output values (y) of a function. In other words, the domain and range represent the horizontal and vertical extent of the function graph, respectively. To find the domain and range of a function, we need to identify the minimum and maximum values of x and y, respectively.
Increasing and Decreasing Intervals
The increasing and decreasing intervals of a function graph represent the intervals where the function is increasing or decreasing, respectively. An increasing interval is where the function outputs are getting larger as the input values increase, while a decreasing interval is where the function outputs are getting smaller as the input values increase.
X and Y Intercepts
The x-intercept is the point where the function graph intersects the x-axis, while the y-intercept is the point where the function graph intersects the y-axis. The x-intercept represents the input value where the function output is zero, while the y-intercept represents the output value where the input value is zero.
Asymptotes
An asymptote is a line that a function approaches but never touches. There are two types of asymptotes: horizontal and vertical. A horizontal asymptote is a line that the function approaches as the input values get larger or smaller, while a vertical asymptote is a line that the function approaches as the input values approach a certain value.
How to Interpret Function Graphs?
Now that we understand the characteristics of function graphs, we can use them to interpret the behavior of a function. For example, we can determine whether a function is increasing, decreasing, or constant by looking at its increasing and decreasing intervals. We can also find the maximum and minimum values of a function by looking at its domain and range. Additionally, we can identify any asymptotes and intercepts to better understand the behavior of the function.
Examples of Function Graphs
Let's take a look at some examples of function graphs and how we can interpret their characteristics:
Example 1: f(x) = x^2
The function graph of f(x) = x^2 is a parabola that opens upwards. Its domain is all real numbers, and its range is all non-negative real numbers. It has no intercepts with the x-axis, but its y-intercept is (0,0). The function is increasing on the intervals (-∞,0) and (0,∞) and decreasing on the interval (0,0). There are no horizontal or vertical asymptotes.
Example 2: g(x) = 1/x
The function graph of g(x) = 1/x is a hyperbola that has two branches. Its domain is all real numbers except for x=0, and its range is all real numbers except for y=0. It has an x-intercept at (1,0) and a y-intercept at (0,1). The function is decreasing on the intervals (-∞,0) and (0,∞). It has a vertical asymptote at x=0 and a horizontal asymptote at y=0.
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