(a) On the same coordinate axes, sketch graphs (as...

(a) On the same coordinate axes, sketch graphs (as accurately as possible) of the functions $y=x^{3}-2 x^{2}-x+2 \quad$ and $\quad y=-x^{2}+5 x+2$ (b) Based on your sketch in part (a), at how many points do the two graphs appear to intersect? (c) Find the coordinates of all intersection points.

(a) On the same coordinate axes, sketch graphs (as accurately as possible) of the functions $y=x^{3}-2 x^{2}-x+2 \quad$ and $\quad y=-x^{2}+5 x+2$
(b) Based on your sketch in part (a), at how many points do the two graphs appear to intersect?
(c) Find the coordinates of all intersection points.

Key Concepts

Polynomial Equation Solving and Factorization

Solving the polynomial equation derived from setting two functions equal can require a variety of techniques such as factoring, using the Rational Root Theorem, synthetic division, or applying the quadratic formula when the equation is reducible to a quadratic form. These methods help identify the roots of the equation, which are crucial for determining the coordinates of the intersection points.

Setting Up and Solving Systems of Equations

When two functions are set equal to each other to find their intersection points, the outcome is a system of equations. Solving this system often involves reducing it to a single polynomial equation whose solutions correspond to the x-coordinates of the intersections. Once these values are determined, they are substituted back into one of the original equations to find the corresponding y-coordinates.

Graphing Polynomial Functions

Graphing polynomial functions involves plotting curves defined by polynomial equations on the coordinate system. Recognizing the degree of the polynomial helps predict the end behavior of the graph, while finding key features such as intercepts, turning points, and points of inflection allows for a more accurate sketch of the function’s overall shape.

Intersection of Graphs

The intersection points of two graphs represent the set of solutions that satisfy both equations simultaneously. In order to determine these points, one typically equates the two functions and solves the resulting equation. Graphical analysis can provide an initial estimate of both the number and the approximate location of these intersections.

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