Math Problem Statement

Graph Set A = {(x, y): (2, -2), (4, -1), (6, 0), . . .}

Solution

To graph Set A, we first recognize that the set consists of points of the form (x,y)(x, y), where the given points are (2,2)(2, -2), (4,1)(4, -1), (6,0)(6, 0), and likely follow a specific pattern.

Looking at the x-values: 2,4,6,2, 4, 6, \dots, they increase by 2 each time. Looking at the y-values: 2,1,0,-2, -1, 0, \dots, they increase by 1 each time.

Thus, the next few points are: (8,1),(10,2),(8, 1), (10, 2), \dots

This suggests a linear relationship between xx and yy.

To find the equation of the line:

  1. Pick two points, say (2,2)(2, -2) and (4,1)(4, -1).
  2. The slope mm is calculated as: m=y2y1x2x1=1(2)42=12m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - (-2)}{4 - 2} = \frac{1}{2}
  3. Using the point-slope form of a line, yy1=m(xx1)y - y_1 = m(x - x_1), and the point (2,2)(2, -2): y(2)=12(x2)y - (-2) = \frac{1}{2}(x - 2)
  4. Simplifying: y+2=12(x2)y=12x3y + 2 = \frac{1}{2}(x - 2) \quad \Rightarrow \quad y = \frac{1}{2}x - 3

Thus, the equation of the line is y=12x3y = \frac{1}{2}x - 3.

Now, to graph Set A, plot the points and draw the line y=12x3y = \frac{1}{2}x - 3.

Would you like me to generate the graph, or do you need further clarification?

Related Questions:

  1. How do you graph points on a coordinate plane?
  2. What is the significance of the slope in the equation of a line?
  3. How do you determine the pattern in a set of points?
  4. Can you explain the point-slope form of a line equation in more detail?
  5. How would the graph change if the slope was negative?

Tip: Always look for patterns in both x- and y-values when identifying a trend in a set of points.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Patterns in Coordinate Sets
Graphing on the Coordinate Plane

Formulas

Slope formula: m = (y2 - y1) / (x2 - x1)
Point-slope form: y - y1 = m(x - x1)
Linear equation: y = mx + b

Theorems

Linear Relationship Theorem
Pattern Recognition in Sequences

Suitable Grade Level

Grades 8-10