Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
![### Understanding the Slope-Intercept Form of a Line
**Objective:** Learn how to write the slope-intercept form of the equation of a line.
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The slope-intercept form of a linear equation is given by:
\[ y = mx + b \]
Where:
- \( y \) is the dependent variable.
- \( x \) is the independent variable.
- \( m \) is the slope of the line.
- \( b \) is the y-intercept of the line.
### Step-by-Step Breakdown:
### 1. Understanding the Graph:
The graph provided shows a blue line on a coordinate plane with a grid. Here are a few key details about the graph:
- The x-axis and y-axis are clearly marked, with both positive and negative values extending to ±5.
- The blue line crosses the y-axis at a certain point, which we'll identify as the y-intercept (b).
- By analyzing the rise over run from the graph, we can determine the slope (m).
### 2. Identifying the Y-Intercept (b):
- The y-intercept is the point where the line crosses the y-axis.
- In this graph, it crosses at the point (0, -1).
- Therefore, \( b = -1 \).
### 3. Determining the Slope (m):
- Slope is calculated as the change in y (vertical) over the change in x (horizontal).
- From the graph, identify two points the line passes through. For instance:
- Point 1: (0, -1), the y-intercept
- Point 2: (1, 1)
#### Calculating Slope:
\[ m = \frac{\text{Change in y}}{\text{Change in x}} = \frac{1 - (-1)}{1 - 0} = \frac{2}{1} = 2 \]
The slope, \( m \), is therefore 2.
### 4. Writing the Equation:
Putting the identified slope and y-intercept into the slope-intercept formula:
\[ y = 2x - 1 \]
### Summary:
The equation of the line in slope-intercept form is:
\[ y = 2x - 1 \]
By understanding how to identify the slope and y-intercept from a graph, you can easily convert any linear equation into its slope-intercept form. Practice analyzing different graphs to become proficient](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fea59d659-13ea-4073-88ed-0121204c1b21%2F8b031fc8-e485-426b-9393-e5f8519822bd%2Fnc0oyu_processed.jpeg&w=3840&q=75)
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