Graphing y = x³ - 4x² - 7x + 10

Algebra 2 · Axiom Academy

EXAMPLE Graphing y = x^3 - 4x^2 - 7x + 10 Complete analysis: finding zeros, multiplicity, end behavior, and sketching the graph. Analyze and graph y = x^3 - 4x^2 - 7x + 10 completely: find every zero (with multiplicity), determine the end behavior, locate the y -intercept, and sketch the graph. The Graph of y = x^3 - 4x^2 - 7x + 10 Sampling the function directly confirms the analysis: the curve crosses the x -axis at each zero, passes through the y -intercept at (0, 10) , and follows the predicted end behavior in both directions. — multiplicity 1 at each, so the curve crosses the axis every time. Falls to the left, rises to the right: as ; as . 2 turning points — the maximum for a degree- 3 polynomial (a local max near , a local min near ). Excellent work — you've completed a full analysis of a cubic polynomial. Here's what we learned: Find zeros systematically: use the Rational Root Theorem to list candidates, test them, then factor completely. Multiplicity matters: odd multiplicity (like 1 ) means the graph crosses the x -axis; even multiplicity means it touches and bounces back. End behavior comes from degree and leading coefficient: odd degree with a positive leading coefficient means the graph falls on the left and rises on the right. Turning points: a degree- n polynomial has at most n-1 turning points — this cubic has exactly 2 . Put it all together: zeros, multiplicity, end behavior, and the y -intercept combine into a complete, accurate sketch.

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