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Precalculus: Parallel and Perpendicular Lines, Average Rate of Change

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Termini in questo insieme (20)

  • What is the slope of any line parallel to y = 3x?
    The slope of any line parallel to y = 3x is \(3\).
  • Write the point-slope form of a line parallel to y = 3x passing through (1, -1).
    Using slope m = 3 and point (1, -1), the point-slope form is
    y + 1 = 3(x - 1).
  • Write the slope-intercept form of the line parallel to y = 3x passing through (1, -1).
    Slope-intercept form is
    y = 3x - 4.
  • What is the slope of a line perpendicular to the vertical line x = 4?
    A vertical line has an undefined slope. A line perpendicular to it is horizontal with slope \(0\).
  • Write the equation of a line perpendicular to x = 4 passing through (8, 2).
    Since slope is 0, the line is horizontal:
    y = 2.
  • How do you find the slope of a line perpendicular to a line with slope m?
    The slope of a perpendicular line is the negative reciprocal of m, i.e., \(-\frac{1}{m}\).
  • Find the slope of a line perpendicular to y = 2x.
    Slope of y = 2x is 2, so perpendicular slope is \(-\frac{1}{2}\).
  • Write the point-slope form of a line perpendicular to y = 2x passing through (4, 3).
    Using slope \(-\frac{1}{2}\) and point (4, 3):
    y - 3 = -\(\frac{1}{2}\)(x - 4).
  • What is the average rate of change of a function between two points?
    The average rate of change is the slope of the secant line between two points:
    \(\frac{f(x_2) - f(x_1)}{x_2 - x_1}\).
  • Find the average rate of change of f(x) = x^2 - 2x from x = 3 to x = 6.
    Calculate:
    (f(6) - f(3)) / (6 - 3) = (24 - 3) / 3 = 7.
  • What is the slope of lines parallel to 2x - 9y - 8 = 0?
    Rewrite as y = (2/9)x - 8/9, so slope is \(\frac{2}{9}\).
  • Write the equation of a line parallel to 2x - 9y - 8 = 0 passing through (-7, 6).
    Using slope \(\frac{2}{9}\) and point (-7, 6):
    2x - 9y + 68 = 0.
  • What is the product of slopes of two perpendicular lines?
    The product of their slopes is \(-1\).
  • Find the equation of a line perpendicular to x - 2y - 3 = 0 passing through (4, -7).
    Slope of given line is \(\frac{1}{2}\), so perpendicular slope is \(-2\). Equation:
    y = -2x + 1.
  • What is the vertical line test used for?
    To determine if a graph represents a function by checking if any vertical line intersects the graph more than once.
  • Define domain and range of a function.
    Domain is the set of all possible input values (x), and range is the set of all possible output values (f(x)).
  • What is the difference quotient formula?
    The difference quotient is
    \(\frac{f(x+h) - f(x)}{h}\), used to find the average rate of change.
  • What characterizes even and odd functions?
    Even functions satisfy f(-x) = f(x); odd functions satisfy f(-x) = -f(x).
  • What is a piecewise function?
    A function defined by different expressions for different intervals of the domain.
  • How do you find the equation of a line given a point and slope?
    Use point-slope form:
    y - y_1 = m(x - x_1), where (x_1, y_1) is the point and m is the slope.