Characteristics of a Polynomial Math Journal by Summer Osak

End behaviour

summero24 — 2022-09-29 19:18:56 UTC

End behaviour describes the trend of the graph of a function as x--> ∞ and as x --> -∞. The potential end behaviours of left and right arms are listed below.

Left arm down: x --> - ∞, y --> - ∞
Left arm up: x --> - ∞, y --> ∞

Right arm down: x --> ∞, y --> - ∞
Right arm up: x --> ∞, y --> ∞

If the degree of a term is even, both arms will be in the same direction and their end behaviours will be the same. For example, if both arms of a particular graph are facing upward, the end behaviours of that graph will be x --> - ∞, y --> ∞ (for the left arm) and x --> ∞, y --> ∞ (for the right arm). However, if the degree of the term is odd, the arms will be in different directions and their end behaviours will be different.

Degree

summero24 — 2022-09-29 19:21:38 UTC

The degree of a term is the total power to which the variables in a given term are raised (i.e. 3x^2 has a degree of 2). The degree of a polynomial equals that of the term with the highest degree in that polynomial (i.e. the degree of y = x^2 + 2x + 3 is 2).

Leading coefficient

summero24 — 2022-10-01 21:35:50 UTC

The leading coefficient --- represented by the letter a --- is the coefficient of the term with the highest degree in a polynomial. It is directly correlated with the right arm's end behaviour.

a > 0: x --> ∞, y --> ∞
a < 0: x --> ∞, y --> - ∞

X - intercepts/ order of x - intercepts/ behaviour of the curve at the x - intercepts

summero24 — 2022-10-01 21:52:22 UTC

The x-intercept is where a graph crosses the x-axis. The order of a root (or an x-intercept) is the number of times that root is repeated. The order of a root is directly correlated with the shape of a graph. For example, if a root has an order of 2, it will be shaped like a parabola on a graph.

Y - intercept

summero24 — 2022-10-01 21:56:36 UTC

The y - intercept is where a graph touches the y - axis.

Turning points

summero24 — 2022-10-01 22:05:33 UTC

A turning point is a point where a graph changes from increasing to decreasing and vice versa. The number of turning points a polynomial has, at most, equals the degree of that polynomial minus 1 (i.e. there are 2 turning points in the graph of y = x^3).

Determining an equation from a graph (Picture 2)

summero24 — 2022-10-02 17:05:37 UTC

My process of determining an equation from a graph of a polynomial is attached above.

Characteristics in Picture 2:

Leading coefficient: 1

Degree: 3

Y-intercept: (0,0)

X - intercepts:

(2,0): the factor is (x-2), the order is 1, will take the shape of a line on the graph.

(0,0): the factor is x, the order is 1, will take the shape of a line on the graph.

(-2,0): the factor is (x+2), the order is 1, will take the shape of a line on the graph.

# of turning points: 2

End behaviour:

left arm: x --> - ∞, y --> - ∞
right arm: x --> ∞, y --> ∞

How to graph y = 2x^2 + 7x + 6 (Picture 1)

summero24 — 2022-10-02 22:26:23 UTC

The graph of y = 2x^2 + 7x + 6 -- as well as an explanation on how to sketch it -- is attached above.

Characteristics of Picture 1:

Leading coefficient: 2

Degree: 2

Y-intercept: (0,6)

X - intercepts:

(-3/2,0): factor is (2x+3), the order is 1, will take the shape of a line on the graph.

(-2,0): factor is (x+2), the order is 1, will take the shape of a line on the graph.

# of turning points: 1

End behaviour:

Left arm: x --> - ∞, y --> ∞
Right arm: x --> ∞, y --> ∞

How to divide x^3 + 7x^2 + 14x + 9 by x + 2 (long division method)

summero24 — 2022-10-03 14:27:03 UTC

An explanation on how to divide x^3 + 7x^2 + 14x + 9 by x + 2 using the long division method is attached above.