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.
![### Determine the Equation of the Graph
Below is a screenshot of a graph depicting a quadratic function. This function creates a parabolic curve that opens upwards.
#### Quadratic Graph Description:
- The vertex of the parabola is located at (0, 1) on the coordinate grid.
- The curve passes through the point (0, 1), indicating that this is the minimum point.
- The parabola appears symmetric across the y-axis.
- The equations of vertical lines drawn through the integers on the x-axis are marked from -3 to 3.
#### Question:
What is the equation of the graph?
Please enter your answer in the box provided.
\[ y = \]
##### Instructions:
1. **Do not use any spaces.**
2. **Use `^` to represent exponents.** For example, write \(2^3\) to indicate \(2^3\).
Considering the vertex form of a quadratic function, which is \( y = a(x-h)^2 + k \), where \((h, k)\) is the vertex of the parabola, determine the equation that fits this graph.
**Example of Input:**
If you believe the function is \( y = 2x^2 \), you should enter:
\[ y = 2x^2 \]
Click "Next page" to proceed after entering your answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4291e50e-f902-4da6-ab90-d00cec455577%2F2e74ef8e-1067-4c2a-9226-748b10173811%2Ftvhanht_processed.jpeg&w=3840&q=75)
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 3 images
