Graph a line that is parallel to the given line. Determine the slope of the given line and the one you graphed in simplest form. Click and drag on the graph to draw a line. Click and drag to plot a parallel line. The line will change colors when a parallel or perpendicular line is drawn accurately. 10 9. 8. 41 3. 10 9 -8 -4 3 2 1 3456 89 s0
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 Slopes of Vertical Lines
In this section, we will explore the properties of the slopes of vertical lines.
#### Graph Explanation:
The image shows a Cartesian coordinate system with the x-axis ranging from -10 to 10 and the y-axis ranging from -10 to 3. There are two vertical lines depicted on the graph:
1. A dark blue vertical line intersecting the x-axis at \( x = -6 \).
2. A light blue vertical line intersecting the x-axis at \( x = 3 \).
Both lines span from the upper part of the graph (just above y=3) down to the lower part (just below y=-10).
#### Concept of Slope:
The slope of a line is a measure of its steepness and direction. It is defined as the change in the y-coordinate (vertical change) divided by the change in the x-coordinate (horizontal change) between two points on the line.
### Interactive Exercise:
Based on the given graph, answer the following questions about the slopes of the provided lines.
1. **Original Slope:**
- Determine the slope of the dark blue vertical line intersecting the x-axis at \( x = -6 \).
\[ Original \text{ Slope: } \boxed{\text{undefined}} \]
2. **Parallel Slope:**
- Determine the slope of the light blue vertical line intersecting the x-axis at \( x = 3 \).
\[ Parallel \text{ Slope: } \boxed{\text{undefined}} \]
*Note: Vertical lines always have an undefined slope because the change in x (horizontal change) is zero, which would make the denominator of the slope calculation zero, resulting in an undefined slope.*
After computing the slopes, click on the "Submit Answer" button to verify your results.
We encourage you to explore more practical applications of slopes to strengthen your understanding of geometric properties in different contexts.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc68938d-169b-4ce7-a57e-c3d38168ed00%2F793471ad-eb11-4317-8a86-aa23b20a4d50%2F7zi9pn_processed.jpeg&w=3840&q=75)
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
