How do I begin to graph this polynomial function?

Question

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior.

P(x) = (2 − x)(x + 3)

Sketch the graph of the polynomial function. Make sure your graph shows all intercepts and exhibits the proper end behavior.

P(x) = −(x + 1)²(x − 1)³(x − 2)

Answer 1

P(x) = (2 − x)(x + 3)

ROOTS:

Find the roots/zeros of the equation:

2-x = 0, so x=2

Plot the point (2,0).

x+3=0, so x=-3

Plot the point (-3,0).

The multiplicity of each of these zeros is odd (1), so the function crosses the x-axis at both points.

Y-INTERCEPT:

Multiply the factors:

P(x)= (2 − x)(x + 3) = -x² -3x +2x + 6 = -x² -x +6

P(0) = 6

Plot the y-intercept: (0,6).

MAXIMUM/MINIMUM

There is only one of these because it is a second degree polynomial (that is, a quadratic or parabola).

It is located at x = -b/a/2 for P(x) = ax² + bx + c.

Plot this point: ( -b/a/2 , P ( -b/a/2) ).

It is a maximum point because the leading coefficient is negative.

END BEHAVIOR:

NOTE that the polynomial has a negative leading coefficient and has an even degree (2).

That means that this polynomial will approach negative infinity as x approaches negative infinity, and approach negative infinity again as x approaches (positive) infinity.

CONNECT THE SEGMENTS SMOOTHLY

Start at negative infinity and connect a curved line to (-3,0). Continue to (0,6) and then to (2,0). From there, head on down toward negative infinity.

Cross the x-axis only at the zeros. Always keep your curve increasing on the x-axis; that is, do not back up!

If you would like a smoother curve, add a point or two between each of these points: Pick an x-value and determine the corresponding y-value. Then plot it and connect smoothly to the points you already have.

Answer 2

(2-x)(x+3)

has 2 x intercepts: x=2 and -3

plot those two points

it's a quadratic function, a parabola, downward opening

rewrite it in vertex form

-x^2-x +6

=-(x^2 +x + 1/4) +6 +1/4

= -(x+1/2)^2 + 25/4

vertex = max = (-1/2,25/4)

plot that point

connect the 3 points with a smooth parabolic curve

(0,6) is another point on the curve

as it's symmetrical about the line of symmetry, x=-1/2

another point is (-1, 6)

that's 5 points, draw a smooth parabolic shaped curve through them all

-(x+1)^2(x-1)^3(x-2)

has x intercepts x=-1,+1,+2

but at x =-1, the graph touches rather than crosses the x axis

the graph is tangent to the x axis when x=-1, the point (-1,0)

end behavior is P(x) approaches +infinity as x approaches - infinity

and approaches - infinity as x approaches + infinity

it's a 6 degree polynomial (2+3+1, add the exponents) so it has 5 turning points