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Let f be given by:

$f:x \to y = \frac{{ - {x^2} + 5x + 8}}{{2{x^2} - 8}}$

a) ${D_f} = \mathbb{R}\backslash \left\{ ? \right\}$

b) Determine the roots.

c) Determine the asymptotes.

d) Sketch the graph.

From our studies of rational functions (block I, learning sequence 2), we know the following

Properties of rational functions

a) Domain

The domain consists of all real numbers for which the denominator is different from 0. Thus, we have to determine the roots of

$2{x^2} - 8 = 0 $ \quad $

which is a quadratic equation, and exclude them from the real numbers.

We rearrange terms by adding 8 to both sides: $\quad 2{x^2}=8$

It makes sense to divide both sides by 2: $\quad {x^2}=4$

We know that there are exactly 2 solutions for this equation, plus 2 and minus two, that is we have

$ {x_1} = -2 \quad {x_2} = 2$.

This answers part a):  ${D_f} = \mathbb{R}\backslash \left\{2,-2 \right\}$

b) Roots

The roots of a rational function coincide with the roots of its numerator term (provided the denominator is unequal zero). Thus we have to determine the solutions of

$ -{x^2} + 5x + 8 = 0$

which again is a quadratic equation. To find the solutions, either make use of the formula from the script or put your calculator to work (programs QUADGL or QUADEQ).

Both approaches lead to

${x_3} = -1.275 \quad {x_4} = 6.275 $

(You can verify that for both of these values, the denominator is unequal zero.)

c) Asymptotes

Vertical asymptotes:

They pass through the roots of the denominator term determined in part a).
Their equations are x=2, and x=-2, respectively.

Horizontal asymptote:

The numerator and the denominator terms are both 2nd degree polynomials:
degree(numerator) = degree(denominator) = 2.

Thus, the quotient of the leading coefficients shows the y-value of the asymptote.

$\frac{{ \color {red} {- 1} \cdot {x^2} + 5x + 8}}{{\color {red} {2} \cdot {x^2} - 8}}\, \longrightarrow \, - \frac{1}{2} \quad $ for ${x \to \infty }$

(So the horizontal asymptote has the equation $y=-\frac{1}{2}$.)

d) Sketch the graph

Start with the roots and the asymptotes.

See diagram 1 to control results

Then, draw a couple of points and link them to sketch the graph.

See diagram 2 to control results

a) Domain

Domain = real numbers except roots of the denominator.

$2{x^2} - 8 = 0 \quad \Rightarrow \quad {x_1} = -2 \quad {x_2} = 2 \quad $


${D_f} = \mathbb{R}\backslash \left\{2,-2 \right\}$

b) Roots

Determine roots of the numerator:

$ -{x^2} + 5x + 8 = 0 \quad \Rightarrow \quad {x_3} = -1.275 \quad {x}_{4} = 6.275 \quad $

c) Asymptotes

Vertical asymptotes:

Roots of the denominator: -2 and 2  (see a).

Horizontal asymptote:

Numerator and denominator have the same degree:

$\frac{{ \color {red} {- 1} \cdot {x^2} + 5x + 8}}{{\color {red} {2} \cdot {x^2} - 8}}\, \longrightarrow \, - \frac{1}{2} \quad $ für ${x \to \infty }$

(Equation of the asymptote: $\;y=-\frac{1}{2}$.)

d) Sketch the graph

See here .