Geometry of Lines and Functions

Geometry of Lines and Functions

Geometry of Lines

  • The equation of a straight line on a graph can be expressed as y = mx + c where:
    • m is the gradient of the line.
    • c is the y-intercept (the point where the line crosses the y-axis).
  • The gradient of a line (m) measures its steepness, and can be calculated using the formula (y2 - y1) / (x2 - x1).
  • If two lines are parallel, their gradients are equal.
  • If two lines are perpendicular, the product of their gradients is -1.

  • A horizontal line has a gradient of 0, while a vertical line has an undefined gradient.

  • The distance between two points (x1, y1) and (x2, y2) can be calculated using the formula sqrt((x2 - x1)² + (y2 - y1)²).
  • The midpoint between two points (x1, y1) and (x2, y2) can be found with the formula ((x1 + x2)/2, (y1 + y2)/2).

Geometry of Functions

  • A function is a rule that assigns to each value in its domain exactly one value in its range.
  • The domain of a function is the complete set of possible values of the independent variable (usually the x-value).

  • The range of a function is the complete set of possible output values (usually the y-value).

  • A quadratic function is a function that can be described by an equation of the form f(x) = ax² + bx + c, where a ≠ 0.

  • A cubic function is a function that can be described by an equation of the form f(x) = ax³ + bx² + cx + d, where a ≠ 0.

  • The inverse of a function switches the roles of the x and y values.

  • The function and its inverse are reflections of each other in the line y = x.

  • A composite function is a combination of two functions, such as f(g(x)).

  • Function transformation: If f(x) is your original function, then:
    • f(x) + a shifts the graph a units vertically.
    • f(x + a) shifts the graph a units horizontally.
    • f(axe) stretches the graph horizontally by a factor of 1/a.
    • af(x) stretches the graph vertically by a factor of a.
  • When sketching curves, consider the points where the curve intersects the axes (x and y intercepts), turning points, regions where the function is increasing or decreasing, asymptotes, and points of inflexion.

Intersections

  • The intersection of two lines is found by solving the equations of the two lines simultaneously.
  • The intersection of a line and a curve is found by substituting the equation of the line into the equation of the curve and solving for the values of x and y.