10 -8 -6 -4 -2 -2 8 10 4 =4 -- -10 Write an equation for the graph in slope-intercept form, i.e., y = mx + 6. to 2.
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 Graphs and Equations
Graphs are visual representations of equations and functions, illustrating how values of one variable are related to values of another. Here, we explore how to interpret and derive equations from linear graphs.
#### Figure Explanation
The provided graph is a Cartesian plane with both x and y axes ranging from -10 to 10. These axes intersect at the origin point, labeled as (0,0), dividing the plane into four quadrants. The distinguishing feature of this graph is a red line extending diagonally through the plane.
#### Linear Equation in Slope-Intercept Form
The standard format for the equation of a straight line is known as the slope-intercept form and is expressed as:
\[ y = mx + b \]
In this formula:
- \( y \) is the dependent variable.
- \( x \) is the independent variable.
- \( m \) represents the slope of the line.
- \( b \) is the y-intercept, which is where the line crosses the y-axis.
#### Determining the Equation from the Graph
To write the equation for the red line in the graph in slope-intercept form, we need to identify both the slope (\( m \)) and the y-intercept (\( b \)).
1. **Y-Intercept (\( b \))**:
- Observing the point where the line intersects the y-axis, we see that this occurs at \( (0, 8) \). Thus, \( b = 8 \).
2. **Slope (\( m \))**:
- The slope is calculated by determining the "rise over run," this is the change in the y-values divided by the change in the x-values between two points on the line.
- Using the points \( (0, 8) \) and \( (2, 0) \):
- \( \text{Rise} = 0 - 8 = -8 \)
- \( \text{Run} = 2 - 0 = 2 \)
- Therefore, \( m = \frac{\text{Rise}}{\text{Run}} = \frac{-8}{2} = -4 \).
Thus, substituting \( m \) and \( b \) into the slope-intercept formula, the equation of the line is:
\[ y = -4x + 8 \]
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**Write an equation for the graph in slope-intercept](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcebbde0b-56df-456f-b45f-057baf1d3a0f%2F4586ab02-72ce-426d-bb4b-d8fc41fc9035%2Frz284rc_processed.png&w=3840&q=75)
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