- Problem
- Detailed Solution
- Summary Solution

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 $

Thus:

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

### 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 .