Curriculum placement
Precalculus. This describes the skill, not a learner estimate.
End behavior of a polynomial
Analyze polynomial end behavior multiplicities and turning behavior
Problem types
- End behavior, crossings and turns of a polynomial
- Match a polynomial graph to its factored form
- Choose the true statement about a graphed cubic
Builds on
- Zeros and factors of a polynomial
- Fundamental theorem of algebra · Useful idea
A degree-n polynomial has n complex zeros counting multiplicity, so its graph crosses or touches the x-axis at most n times, and its nonreal zeros come in conjugate pairs that never appear as x-intercepts.
- Designs that meet geometric constraints · Useful idea
The volume of a box cut from a sheet is a cubic in the cut size, whose graph shows the greatest volume.
- The minimum of an upward-opening quadratic · Earlier idea
The minimum of a parabola is a turning point, and higher degree polynomials may have several turning points.
Leads to
- Reading slope and intercept in context · See it applied
Choosing a polynomial to fit nonlinear data uses its end behavior and number of turns to judge which degree fits.
Related skills
- Asymptotes, holes, and intercepts · Related idea
A polynomial's degree and leading coefficient set its end behavior. A rational function compares the degrees and leading coefficients of its numerator and denominator to find its horizontal asymptote.
- Linear growth against exponential growth · Related idea
A polynomial grows slower than any exponential in the long run, which growth comparison shows.
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