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    • ▾Precalc
      • ▸Conic sections
        • •Intro to conic sections
        • •Center and radii of an ellipse
        • •Foci of an ellipse
        • •Intro to hyperbolas
        • •Foci of a hyperbola
        • •Maximizing the product of two integers, given their integer sum
        • •Hyperbolas not centered at the origin
        • •Standard equation of a circle
      • ▸Functions and relations
        • •Composition of functions
        • •Operations with functions
        • •Graphing the sum of trig functions from their graphs
        • •Finding and verifying the inverse of a nonlinear function
        • •Graphically finding the increasing, decreasing, and constant intervals
        • •Vertical and horizontal line tests
        • •Restrict the domain then find the inverse
        • •Sketching the inverse of a function from its graph
      • ▾Characteristics of functions
        • •Finding extrema graphically
        • •End behavior
      • ▸Rational functions
        • •Partial fraction decomposition
        • •Simplifying the difference quotient
        • •Pick's theorem
      • ▸Sigma notation and the binomial theorem
        • •Converting sums between sigma notation and expanded form
        • •Properties of sigma notation
        • •Using the binomial theorem
      • ▸Series
        • •Geometric series
        • •Geometric series with summation notation
        • •Arithmetic series with summation notation
      • ▸Logic
        • •De Morgan's laws for propositional logic
     › Precalc › Characteristics of functions

    End behavior

    Students will learn how to determine the end behavior of univariate polynomials.

    Intro to end behavior of polynomials by Khan Academy

    End behavior of polynomials by Khan Academy

    Conclude by giving your students these challenges:

    • Time to Evolve by NRICH
    • 2007 AMC 8, Problem 6
    • A Dart Probability

    In the figure below, \(C\) is the center of the circle, \(\overline{AE} = 1,\) and \(\overline{EF} = 2.\) What's the length of \(\overline{EB}?\)

    Here's the solution: First, notice that \(\overline{AC} = 3.\) Next, find that \(\overline{AD} = \sqrt{3}\) using the Pythagorean theorem. Notice that \(\triangle ACD\) is a 30-60-90 triangle, with \(\angle DCA = 60^\circ.\) Also, notice that \(ECD\) is an isosceles triangle, because two of its sides are radii of the circle. We already know one of its angles, so the other two can be determined. It happens that \(\triangle ECD\) is equilateral. Thus, \(\overline{EB}\) is half the radius of the circle, so \(\overline{EB} = 1/2.\)

    Suppose we're in a world with natural numbers only. Because we don't have negative numbers, an operation like x - y isn't defined for all possible choices of x and y. For example, 0 - 1 has no sensible value. But subtraction is such a useful operation, we might as well have an operation that behaves something like integer subtraction. The following is such an operation. It's often referred to as "proper subtraction." $$x \mathop{\unicode{8760}} y = \begin{cases} 0 & \text{if }x \lt y \\ x - y & \text{otherwise} \end{cases}$$

    That takes care of subtraction, but what about division? We don't have rationals, so 1/2 doesn't make any sense. To handle division, we're going to define what's often called "integer division." Our definition appears below. $$\text{div}(x, y) = \left\lfloor \dfrac{x}{y} \right\rfloor$$

    How can we express \(\text{max}\) and \(\min,\) in terms of only \(+,\) \(\unicode{8760},\) \(\times,\) \(\text{div}?\)

    Additional lessons and practice problems