Graph Lines and Equations Study Notes

Graph Lines and Equations

Basic Functions and Linear Equations

• Linear Functions:

  • Format: f(x) = mx + b, where:

  • m = slope of the line

  • b = y-intercept

Examples of Linear Functions:

• f(x) = 3x - 5:

  • Slope (m) = 3

  • Y-intercept (b) = -5

• f(x) = -2x + 4:

  • Slope (m) = -2

  • Y-intercept (b) = 4

• f(x) = x + 3:

  • Slope (m) = 1

  • Y-intercept (b) = 3

• f(x) = 5x - 2:

  • Slope (m) = 5

  • Y-intercept (b) = -2

Graphing Linear Equations

• Standard Form of Linear Equations:

  • General Form: $Ax + By = C$

  • Where A, B, and C are constants.

Example Equations:

• 2x + 5y = 20:

  • To find y:

  • Rearrange: $5y = -2x + 20$

  • Result: $y = - rac{2}{5}x + 4$

• 3x - 4y = 12:

  • To find y:

  • Rearrange: $-4y = -3x + 12$

  • Result: $y = rac{3}{4}x - 3$

More Linear Equations

• 2x - 3y = 12:

  • Rearranged: $-3y = -2x + 12$

  • Result: $y = rac{2}{3}x - 4$

• x + 3y = -6:

  • Rearranged: $3y = -x - 6$

  • Result: $y = - rac{1}{3}x - 2$

• x - y = 5:

  • Rearranged: $-y = -x + 5$

  • Result: $y = x - 5$

• 2x - y = 4:

  • Rearranged: $-y = -2x + 4$

  • Result: $y = 2x - 4$

More Functions and Their Forms

Non-Linear Functions:

• f(x) = rac{1}{1}

• f(x) = x + 2

Specific Points on Graphs

• Vertical Lines:

  • Example: x = -3

  • Represents a vertical line at x = -3

  • Example: x = 5/2 (or 2.5)

  • Represents a vertical line at x = 2.5

Exercise: Write the equations of following graphs

• Task: Students should identify patterns in the graphs and formulate the corresponding equations.

Additional Practice Problems

• Write equations for the following scenarios:

  • Example forms:

  • Y = mx + b

  • Y = C (with horizontal lines)

  • X = C (with vertical lines)

Conclusion

• The notes compiled provide comprehensive coverage of concepts related to graph lines and linear equations, including real examples and their calculations. This serves as a functional guide for understanding graph representations in mathematics.