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.
![## Graphing a Line through a Given Point with a Given Slope
**Objective:** Learn how to graph a line with a given slope passing through a specific point.
### Example Problem
**Graph the line with slope \(\frac{2}{3}\) passing through the point (2, 4).**
#### Instructions:
1. **Identify the slope (m):** In this example, the slope (m) is \(\frac{2}{3}\). This means that for each increase of 3 units in the x-direction, there is an increase of 2 units in the y-direction.
2. **Identify the point (x₁, y₁):** The line passes through the point (2, 4).
3. **Plot the given point on the coordinate plane:** Start by plotting the point (2, 4).
4. **Use the slope to find another point on the line:**
- From (2, 4), move 3 units to the right (x-direction) to reach x = 5.
- Then move 2 units up (y-direction) to reach y = 6.
- Plot the new point (5, 6).
5. **Draw the line:** Using a ruler or line-drawing tool, draw a line through the two points: (2, 4) and (5, 6).
#### Diagram Explanation:
- The graph is on a standard Cartesian plane with x and y-axes labeled from -10 to 10.
- The first point (2, 4) is marked.
- From this point, a line follows the slope \(\frac{2}{3}\), intersecting at (5, 6).
- The line is drawn, extending in both directions.
#### Tools Provided:
- **Pencil Icon**: To draw points and lines.
- **Eraser Icon**: To remove drawn points or lines.
- **Undo Icon**: To undo the last action.
- **Help Icon**: For additional hints or guidance.
### Explanation Button:
- **Check Button:** Verifies if the drawn line correctly represents the given slope and point.
### Additional Assistance:
If you need further explanation, click the **Explanation** button for step-by-step guidance.
By following these steps, you should be able to graph any line given its slope and a point it passes through. Happy graphing!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F96b8c0f1-94d8-4529-9908-c3bb8a31dc84%2F76c8f2e2-b25e-4a57-805f-aa5a9ca95b69%2F89thc89_processed.jpeg&w=3840&q=75)
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