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Precalculus Unit 2 Study Guide Flashcards

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Terms in this set (25)

  • Library of Functions
    A set of basic functions including linear, quadratic, cubic, absolute value, square root, and rational functions used as building blocks for more complex functions.
  • Piecewise Function
    A function defined by different expressions for different intervals of the domain, written using correct notation with cases.
  • Example of Piecewise Function
    f(x) = { 2x if x < -1; 0 if x ≥ -1 }
  • Transformations of Functions
    Changes to the graph of a function including vertical/horizontal shifts, reflections, stretches, and compressions.
  • Vertical Stretch by Factor of 2
    Multiply the function by 2: if f(x) = 1/x, then the stretched function is g(x) = 2 * (1/x) = 2/x.
  • Reflection over y-axis
    Replace x by -x in the function: if f(x) = x^3, then reflected function is g(x) = (-x)^3 = -x^3.
  • Linear Function Slope and Intercept
    Slope is the rate of change; y-intercept is the value when x=0. For y = -1/3x - 3, slope = -1/3, y-intercept = -3.
  • Average Rate of Change
    Change in function values divided by change in x over an interval; indicates if function is increasing, decreasing, or constant.
  • Linear Model Determination
    A function is linear if the rate of change is constant across data points.
  • Solving Linear Inequalities
    Find values of x that satisfy inequalities like f(x) > g(x) by solving algebraically.
  • Cost Function Example
    C(x) = 3x + 60 models bike rental cost; 3 is hourly rate, 60 is fixed fee.
  • Domain of Cost Function
    All x ≥ 0 since rental hours cannot be negative.
  • Quadratic Equation Solving Methods
    Use factoring, completing the square, or quadratic formula depending on the equation.
  • Convert Quadratic to Vertex Form
    Rewrite f(x) = ax^2 + bx + c as f(x) = a(x - h)^2 + k where (h,k) is the vertex.
  • Vertex of Quadratic Function
    Point (h,k) where the parabola changes direction; axis of symmetry is x = h.
  • Concavity of Quadratic Graph
    If a > 0, graph is concave up (minimum vertex); if a < 0, concave down (maximum vertex).
  • Discriminant and Number of Roots
    Discriminant D = b^2 - 4ac determines roots: D > 0 two real roots, D = 0 one real root, D < 0 no real roots.
  • Finding x- and y-intercepts of Quadratic
    x-intercepts: solve f(x) = 0; y-intercept: evaluate f(0).
  • Domain and Range of Quadratic
    Domain is all real numbers; range depends on vertex and concavity.
  • Quadratic Word Problem Setup
    Translate real-world constraints into quadratic equations to solve for maximum or minimum values.
  • Maximizing Area with Fixed Perimeter
    Use quadratic functions to find dimensions that maximize area given fencing length.
  • Revenue Function from Demand Equation
    Revenue R(p) = price × quantity; often quadratic, used to find max revenue and optimal price.
  • Projectile Motion Quadratic Model
    Height h(t) = -4.9t^2 + vt models vertical motion; domain and range relate to time and max height.
  • Maximum Height of Projectile
    Occurs at vertex of parabola; time found by -b/(2a) in h(t) = at^2 + bt + c.
  • Time When Projectile Lands
    Solve h(t) = 0 for positive t to find when object returns to ground.