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Precalculus Test 2 Key Concepts

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Terms in this set (20)

  • Distance formula between two points
    The distance between points (x₁, y₁) and (x₂, y₂) is \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).
  • Midpoint formula between two points
    The midpoint of segment with endpoints (x₁, y₁) and (x₂, y₂) is \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\).
  • Finding the second endpoint given one endpoint and midpoint
    Use midpoint formula: if midpoint M and endpoint A are known, then second endpoint B is \((2x_M - x_A, 2y_M - y_A)\).
  • Standard form equation of a circle
    A circle with center (h, k) and radius r has equation \((x - h)^2 + (y - k)^2 = r^2\).
  • General form equation of a circle
    The general form is \(x^2 + y^2 + Dx + Ey + F = 0\), where D, E, and F are constants.
  • How to graph a circle from its equation
    Identify center (h, k) and radius r from standard form, then plot center and draw circle with radius r.
  • Determining if a relation is a function
    A relation is a function if each input (x-value) has exactly one output (y-value).
  • Domain of a function
    The set of all possible input values (x-values) for which the function is defined.
  • Range of a function
    The set of all possible output values (y-values) of the function.
  • Evaluating a function at a given input
    Substitute the input value into the function's formula and simplify to find the output.
  • Graphing a line from an equation
    Rewrite the equation in slope-intercept form \(y = mx + b\), plot the y-intercept, then use slope to find another point.
  • Writing the equation of a line given slope and point
    Use point-slope form: \(y - y_1 = m(x - x_1)\).
  • Evaluating a composition of functions
    To find (f ∘ g)(x), first evaluate g(x), then substitute that result into f.
  • Transformations of graphs
    Include translations (shifts), reflections, stretches, and compressions applied to the parent function.
  • Effect of adding a constant outside the function
    Adding a constant k outside f(x) shifts the graph vertically by k units.
  • Effect of adding a constant inside the function argument
    Adding a constant h inside f(x + h) shifts the graph horizontally by -h units.
  • Reflection over the x-axis
    Multiply the function by -1: graph of y = -f(x) is the reflection of y = f(x) over the x-axis.
  • Reflection over the y-axis
    Replace x by -x: graph of y = f(-x) is the reflection of y = f(x) over the y-axis.
  • Vertical stretch or compression
    Multiply f(x) by a factor a: if |a| > 1, vertical stretch; if 0 < |a| < 1, vertical compression.
  • Horizontal stretch or compression
    Replace x by bx: if |b| > 1, horizontal compression; if 0 < |b| < 1, horizontal stretch.