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Precalculus: Linear Functions, Equations, and Inequalities

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Termini in questo insieme (19)

  • What is the general form of a linear function?
    A linear function is generally written as \(y=mx+b\), where m is the slope and b is the y-intercept.
  • How do you find the slope of a line given two points?
    The slope m is calculated by \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
  • What does the slope of a line represent?
    The slope represents the rate of change or how much y changes for a unit change in x.
  • How do you write the equation of a line in point-slope form?
    Point-slope form is \(y - y_1 = m(x - x_1)\), where m is slope and (x_1, y_1) is a point on the line.
  • What is the slope-intercept form of a linear equation?
    Slope-intercept form is \(y = mx + b\), where m is slope and b is the y-intercept.
  • How do you find the y-intercept of a line?
    The y-intercept is the point where the line crosses the y-axis, found by setting \(x=0\) in the equation.
  • What is the standard form of a linear equation?
    Standard form is \(Ax + By = C\), where A, B, and C are integers and A ≥ 0.
  • How do you solve a linear equation?
    Isolate the variable by performing inverse operations to both sides until the variable is alone.
  • What is a linear inequality?
    A linear inequality is like a linear equation but uses <, >, ≤, or ≥ instead of =.
  • How do you graph a linear inequality?
    Graph the boundary line (solid for ≤ or ≥, dashed for < or >), then shade the region that satisfies the inequality.
  • What happens to the inequality when you multiply or divide by a negative number?
    You must reverse the inequality sign when multiplying or dividing both sides by a negative number.
  • How do you determine if a point satisfies a linear inequality?
    Substitute the point's coordinates into the inequality; if true, the point satisfies the inequality.
  • What is the relationship between parallel lines and their slopes?
    Parallel lines have equal slopes but different y-intercepts.
  • What is the relationship between perpendicular lines and their slopes?
    Perpendicular lines have slopes that are negative reciprocals of each other.
  • How do you write the equation of a line parallel to a given line through a point?
    Use the slope of the given line and the point in point-slope form: \(y - y_1 = m(x - x_1)\).
  • How do you write the equation of a line perpendicular to a given line through a point?
    Use the negative reciprocal of the given line's slope and the point in point-slope form.
  • What is the intercept form of a linear equation?
    Intercept form is \(\frac{x}{a} + \frac{y}{b} = 1\), where a and b are x- and y-intercepts.
  • How do you convert from standard form to slope-intercept form?
    Solve for y: \(y = -\frac{A}{B}x + \frac{C}{B}\).
  • What is the solution set of a linear inequality?
    The solution set is all points in the coordinate plane that make the inequality true.