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Precalculus: Lines and Slopes

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  • What is the formula to find the slope of a line given two points?
    The slope m is \(\frac{y_2 - y_1}{x_2 - x_1}\), where (x_1, y_1) and (x_2, y_2) are the two points.
  • What is the point-slope form of a line?
    The point-slope form is \(y - y_1 = m(x - x_1)\), where m is the slope and (x_1, y_1) is a point on the line.
  • What is the slope-intercept form of a line?
    The slope-intercept form is \(y = mx + b\), where m is the slope and b is the y-intercept.
  • What is the general form of a line?
    The general form is \(Ax + By + C = 0\), where A, B, and C are integers and A ≥ 0.
  • What is the equation of a horizontal line?
    A horizontal line has slope 0 and equation \(y = b\), where b is the y-intercept.
  • What is the equation of a vertical line?
    A vertical line has an undefined slope and equation \(x = a\), where a is the x-intercept.
  • How do you graph a line using slope and y-intercept?
    1. Plot the y-intercept (0, b). 2. Use the slope m as rise over run to find a second point. 3. Draw a line through the points with arrows at ends.
  • How do you find the x-intercept of a line in general form?
    Set y = 0 in \(Ax + By + C = 0\) and solve for x.
  • How do you find the y-intercept of a line in general form?
    Set x = 0 in \(Ax + By + C = 0\) and solve for y.
  • What is the slope of a line that rises from left to right?
    The slope is positive.
  • What is the slope of a line that falls from left to right?
    The slope is negative.
  • What is the slope of a horizontal line?
    The slope is zero.
  • What is the slope of a vertical line?
    The slope is undefined.
  • How do you write the equation of a line given slope m and point (x_1, y_1)?
    Use point-slope form: \(y - y_1 = m(x - x_1)\).
  • How do you convert point-slope form to slope-intercept form?
    Solve for y to get \(y = mx + b\), where b is the y-intercept.
  • What does the slope represent geometrically?
    Slope measures the steepness of a line, calculated as rise over run.
  • What is the significance of the y-intercept in a line's equation?
    The y-intercept is the point where the line crosses the y-axis (x=0).
  • What is the significance of the x-intercept in a line's equation?
    The x-intercept is the point where the line crosses the x-axis (y=0).
  • How do you graph a line using intercepts from the general form?
    Find x- and y-intercepts, plot them, then draw a line through these points with arrows at ends.
  • What is the slope of the line passing through points (2,1) and (3,4)?
    Slope m = \(\frac{4 - 1}{3 - 2} = 3\).