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.
![**Task: Write an absolute value equation for the graph shown.**
**Equation:**
\[ y = \_\_ \] (Simplify your answer.)
**Graph Description:**
The graph is plotted on a standard Cartesian coordinate plane with both the x-axis and y-axis. The graph depicts a V-shaped absolute value function. The vertex of the V is located at the point \((0, -4)\).
- The left arm of the V extends upward with a slope of 1, starting from the vertex \((0, -4)\) and passes through points such as \((-4, 0)\).
- The right arm of the V also extends upward with a slope of -1, starting from the vertex \((0, -4)\) and passes through points such as \((4, 0)\).
The graph is symmetric about the y-axis. The V-shape indicates an absolute value function of the form \(y = a|x - h| + k\), where \((h, k)\) is the vertex of the graph. In this case, the vertex \((0, -4)\) suggests that the equation of the absolute value function could be \( y = a|x| - 4\). The symmetry and slopes confirm that \( a = 1 \), leading to the equation \( y = |x| - 4\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa7d34ce-a17c-44ce-a3ca-71da2b3b3b33%2F3cea03a6-fa6b-431e-afa5-1af9fe2a8655%2Fefxenak_processed.png&w=3840&q=75)
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