Rational Functions Math Journal by Summer Osak

X - intercept

summero24 — 2022-10-22 18:46:10 UTC

The x-intercept is where a graph crosses the x-axis. The x-intercept(s) of a rational function can be found by making the numerator equal to 0.

If the order of an x-intercept is odd, there is no turning point at that intercept. However, if the order of an x-intercept is even, there is a turning point at that intercept.

Y - intercept

summero24 — 2022-10-22 18:47:03 UTC

The y - intercept is where a graph crosses the y - axis. The y - intercept of a rational function can be found by making x equal to 0.

Vertical Asymptote

summero24 — 2022-10-22 18:58:39 UTC

A vertical asymptote is a vertical line that a function approaches but never intersects. To find the vertical asymptote of a rational function, you need to make the denominator of the function equal to 0.

If the order of a vertical asymptote is odd, the arms of the function that are closest to that vertical asymptote will go in different directions (in other words, one will face upwards and the other will face downwards). However, if the order of a vertical asymptote is even, the arms that are closest to that asymptote will go in the same direction.

Horizontal Asymptote

summero24 — 2022-10-22 19:02:31 UTC

A horizontal asymptote is a horizontal line that a function approaches but never touches.

If the degree of the numerator of a rational function is greater than that of the denominator, the horizontal asymptote is y = 0. If the degree of the numerator is equal to that of the denominator, the horizontal asymptote is the quotient of the leading coefficient of the numerator and the leading coefficient of the denominator (i.e. if this quotient is 2, the horizontal asymptote is y=2). Finally, if the degree of the numerator is bigger than that of the denominator, there is no horizontal asymptote

Holes

summero24 — 2022-10-22 19:59:09 UTC

In the context of rational functions, a hole is a coordinate on a graph that is undefined (this point is not on a vertical, horizontal, or oblique asymptote). Holes only exist in functions where the numerator and denominator have a common factor

In order to find a hole, you first need to find this common factor (i.e. if the common factor of a function is (x-2), the x value of the hole is 2.) Secondly, in order to find the hole's y value, you need to find the simplified version of the function and make x equal to this x - value (i.e. if the simplified form of the above function is (x-3)(x-4)/(x-5)(x-6), you need to make x = 2)

Oblique Asymptote

summero24 — 2022-10-22 20:00:45 UTC

An oblique asymptote is a line a function approaches but doesn't touch. An oblique asymptote can only exist in rational functions where the degree of the numerator is one larger than that of the denominator (therefore, a rational function cannot have a horizontal and oblique asymptote at the same time.)

In order to find an oblique asymptote, you need to divide the denominator by the numerator.

Leading Coefficient

summero24 — 2022-10-22 20:59:26 UTC

The leading coefficient (or LC) of a rational function is the quotient of the leading coefficient of the numerator and the leading coefficient of the denominator. It's sign is directly correlated with the right arm's end behaviour.

LC < 0: x --> - ∞, y --> the horizontal or oblique asymptote of the function.

LC > 0: x --> ∞, y --> the horizontal or oblique asymptote of the function.

How to Graph y= x^4 - 3x^3 - 7x^2 +15x + 18 / x^4 + 7x^3 + 6x^2 -32x -32

summero24 — 2022-10-24 23:55:04 UTC

The graph of y= x^4 - 3x^3 - 7x^2 +15x + 18 / x^4 + 7x^3 + 6x^2 -32x -32 -- and an explanation on how to sketch it -- is attached above.

Characteristics of the above function:

A hole exists at (-1, 0.6)

X - intercepts:

(-2,0): The factor is (x+2), the order is odd and there is no turning point at this x-intercept

(3,0): The factor is (x-3), the order is even and there is a turning point at this x-intercept

Y - intercept: (0, -0.56)

Vertical asymptotes:

x = 2: The factor is (x-2), the order is odd and the arms closest to this asymptote go in different directions

x = -4: The factor is (x+4), the order is even and the arms closest to this asymptote go in the same direction

Horizontal asymptote: y = 1

End behaviour:

Right arm: x --> ∞, y --> y = 1 (since the LC > 0)
Left arm: x --> - ∞, y --> y = 1

Additional points:

(4, 0.05)
( -0.72, 0.61)