Adaptation of Mathematical ALGorithms

Analysis of the function graph usually consists of the following particular steps which are connected consecutively:

1. Determination of the domain of definition, points of discontinuity.
2. Deciding whether the function is odd, even, or periodical.
3. Solving the equation $f(x) = 0$, describing intervals where the graph is above or below the axis $x$.
4. Solving the equation $f'(x) = 0$, determination of stationary points and intervals where the graph is increasing or decreasing.
5. Solving the equation $f''(x) = 0$, determination of inflection points and intervals where the graph concaves downward or upward.
6. Determination of vertical or oblique asymptotes.
7. Sketching the graf of the function.

We described the steps of the procedure how to analyse the function's graph. To determine the function's properties during the steps 1–6 a blind student needs an accessible mathematical editor in which he performs computations linearly.

In case of the steps 3–5 he/she prepares tables of intervals describing whether the graph of the function

• step 3: is above or below the axis $x$,
• step 4: is increasing or decreasing,
• step 5: concaves downward or upward.

There's a change in the tables' arrangement. A blind student writes down specific symbols indicating the function's properties and uses separate lines for every interval.

In the next part of the site we present a concrete example of the function $f(x) = x^3/(x^2-1)$ and the graph's analysis done by one of our blind students. There's a key issue how to adapt the 7th task – sketching the graph after computing all the necessary properties of the function. We will focus on the problem in the part Sketching the graph of the function and its adaptation.

Example

The example is available in following formats:

1. this HTML page
2. file in Lambda: analysis_en.lambda
3. file in MS Word: analysis_en.doc

Analyse the function $ f(x) =\frac{ x^3 }{ x^2 -1 } $.

Solution

1. Domain of the function

$D_f =(-\infty, -1) \cup (-1, 1) \cup (1, \infty)$
$-1$ and $1$ are the points of discontinuity.

2. Even or odd function

$f(-x) =\frac{ (-x)^3}{ (-x)^2 -1}=-\frac{ x^3}{ x^2 -1} =-f(x)$

The function is odd.

3. Graph of the function below or above the axis $x$

Solving the equation $f(x) =0$:

$$\frac{ x^3}{ x^2 -1} =0 \Leftrightarrow x =0$$
$x \in (-\infty, -1)$: $-$
$x \in (-1, 0)$: $+$
$x \in (0, 1)$: $-$
$x \in (1, \infty)$: $-$

4. Stationary points

Solving the equation $f'(x) =0$:

$f'(x) =\frac{ 3x^2 *(x^2 -1) -x^3 *2x}{ (x^2 -1)^2}$
$=\frac{ x^4 -3x^2}{ (x^2 -1)^2}$
$=\frac{ x^2 *(x^2 -3)}{ (x^2 -1)^2}$

$x =0$
$x^2 -3 =0$
$x^2 =3$
$x =\pm\sqrt{ 3}$

$x \in (-\infty, -\sqrt{ 3})$: +, $\uparrow$
$x \in (-\sqrt{ 3}, 0)$: -, $\downarrow$
$x \in (0, \sqrt{ 3})$: -, $\downarrow$
$x \in (-\infty, -\sqrt{ 3})$: +, $\uparrow$

The local maximum: $[-\sqrt{ 3}, \frac{ -3\sqrt{ 3}}{ 2}]$
The local minimum: $[\sqrt{ 3}, \frac{ 3\sqrt{ 3}}{ 2}]$

5. Inflection points

Solving the equation $f''(x) =0$:

$f''(x) =\frac{ (4x^3 -6x) *(x^2 -1)^2 -(x^4 -3x^2) *2(x^2 -1) *2x}{ (x^2 -1)^4}$
$=\frac{ 2x *(x^2 -1) *[(2x^2 -3) *(x^2 -1) -2(x^4 -3x^2)]}{ (x^2 -1)^4}$
$=\frac{ 2x *[(2x^2 -3) *(x^2 -1) -2(x^4 -3x^2)]}{ (x^2 -1)^3}$
$=\frac{ 2x *[2x^4 -3x^2 -2x^2 +3 -2x^4 +6x^2]}{ (x^2 -1)^3}$
$=\frac{ 2x *[x^2 +3]}{ (x^2 -1)^3}$

$2x =0 \Rightarrow x =0$
$x^2 +3\neq0$

$x \in (-\infty, -1)$: $-$, A
$x \in (-1, 0)$: $+$, V
$x \in (0, 1)$: $-$, A
$x \in (1, \infty)$: $+$, V

The inflection point is $[0, 0]$.

6. Asymptotes

Oblique asymptote $y =kx +q$:

$k =\lim_{ x \rightarrow \pm\infty}\frac{ f(x)}{ x}$
$=\lim_{ x \rightarrow \pm\infty}\frac{ x^3}{ x^2 -1} :x$
$=\lim_{ x \rightarrow \pm\infty}\frac{ x^3}{ x *(x^2 -1)}$
$=\lim_{ x \rightarrow \pm\infty}\frac{ x^2}{ x^2 -1}$
$=1$

$q =\lim_{ x \rightarrow \pm\infty}(f(x) -kx)$
$=\lim_{ x \rightarrow \pm\infty}(\frac{ x^3}{ x^2 -1} -x)$
$=\lim_{ x \rightarrow \pm\infty}(\frac{ x^3 -x(x^2 -1)}{ x^2 -1})$
$=\lim_{ x \rightarrow \pm\infty}\frac{ x}{ x^2 -1}$
$=0$

So the oblique asymptote is $y =x$.

Vertical asymptotes:

• $\lim_{ x \rightarrow -1^-}\frac{ x^3}{ x^2 -1}=\lim_{ x \rightarrow -1^-}\frac{ x^3}{ x +1} *\frac{ 1}{ x -1} =-\infty$
• $\lim_{ x \rightarrow -1^+}\frac{ x^3}{ x^2 -1}=\lim_{ x \rightarrow -1^+}\frac{ x^3}{ x +1} *\frac{ 1}{ x -1} =\infty$
• $\lim_{ x \rightarrow 1^-}\frac{ x^3}{ x^2 -1}=\lim_{ x \rightarrow 1^-}\frac{ x^3}{ x +1} *\frac{ 1}{ x -1} =-\infty$
• $\lim_{ x \rightarrow 1^+}\frac{ x^3}{ x^2 -1}=\lim_{ x \rightarrow 1^+}\frac{ x^3}{ x +1} *\frac{ 1}{ x -1} =\infty$

There are two vertical asymptotes: $x =-1$ and $x =1$.

7. Final description of the graph

The function is defined for $ x \in (-\infty, -1) \cup (-1, 1) \cup (1, \infty)$. The graph is limited by the vertical asymptotes $ x =-1$ and $ x =1$ and the oblique asymptote $ y =x$. There are two local extrema, local maximum: $ [-\sqrt{ 3}, \frac{ -3\sqrt{ 3}}{ 2}]$ and local minimum: $ [\sqrt{ 3}, \frac{ 3\sqrt{ 3}}{ 2}]$.

The graph of the function is for $ x \in (-\infty, -1)$ below the axis $ x$, there is the local maximum $ [-\sqrt{ 3}, \frac{ -3\sqrt{ 3}}{ 2}]$, the curve concaves downward, and its position is on the left side of the asymptote $ x =-1$. Because the point $ [-\sqrt{ 3}, -\sqrt{ 3}]$ lying on the asymptote $ y =x$ is above the local maximum $ [-\sqrt{ 3}, \frac{ -3\sqrt{ 3}}{ 2}]$, the whole part of the graph is for $ x \in (-\infty, -1)$ bellow the asymptote $ y =x$.

The graph of the function on the interval $ (-1, 1)$ intersects with the axis $ x$ at the point $ [0, 0]$ – the inflection point, where the function changes its shape, for $ x \in (-1, 0)$ concaves upward, for $ x \in (0, 1)$ concaves downward, the function is odd. The part of the graph for $ x \in (-1, 1)$ is located between the vertical asymptotes $ x =-1$ and $ x =1$, the oblique asymptote $ y =x$ intersects the graph of the function at the point $ [0, 0]$.

The graph of the function is for $ x \in (1, \infty)$ above the axis $ x$, there is the local minimum $ [\sqrt{ 3}, \frac{ 3\sqrt{ 3}}{ 2}]$, the curve concaves upward, and its position is on the right side of the asymptote $ x =1$. Because the point $ [\sqrt{ 3}, \sqrt{ 3}]$ lying on the asymptote $ y =x$ is below the local minimum $ [\sqrt{ 3}, \frac{ 3\sqrt{ 3}}{ 2}]$, the whole part of the graph is for $ x \in (1, \infty)$ above the asymptote $ y =x$.

The tradional last part of the function analysis is to sketch the graph. Students prove their comprehension to the meaning of all the data they counted before and how they are able to put these results together. Such a task is much more complicated for blind people. Three types of adaptions are considered:

1. handmade production of the graph's tactile version using suitable assistive technology
2. text description of the graph which helps a reader to imagine how it looks visually
3. selection from several tactile images of graphs prepared in advance

We discussed pros and cons of all the three adaptions with participants of our workshop held during ICCHP Summer University 2012 in Linz. The handmade production of the function's tactile image is of course time consuming for blind people. They make much more effort than their sighted peers to ensure the result of their work to be corresponding with the reality.

We can confirm from our own experience the second method of adaptation is not very easy for a blind person. A long-term traning is needed to ensure the blind student is able to describe clearly the graph of the function whose properties he/she counted before. He/she often makes mistakes in ordering information when describing the graph. At first, he/she focuses on details and afterwards adds general information. To have a perfect picture of the graph in his/her mind the blind student needs to connect his/her description with a tactile image. One of the workshop's participants, a teacher of mathematics for blind confirmed that: "I think a blind student misses something if he only describes the graph and doesn't have its tactile version. The text description of the graph is not a hundred per cent replacement of its image."

It can be very easy to supervise the correctness of the graph's description. The blind student provides a teacher or a classmate with the description and asks him/her to sketch the graph without telling him/her the general form of the function. If the drawing corresponds to the real image of the function, the blind student's description is correct. We performed such a test with participants of several workshops aimed at educational staff. We asked them to draw the graph when having its text description (done by the blind student we have shown in the previous part). The result confirms the description is created well. All the information are structured properly and teachers don't have problems to understand it. We can share some interesting observations of teachers who performed the test. They missed few details which we as the teachers of the blind student had not noticed. For example, the description doesn't contain the computation of the first derivative in the inflection point which indicates how the function's graph changes at the neighborhood of that point, how much it is close to the axis $x$.

There are several factors which affect the difficulty of the task when using the third method of adaptation. At first, the more tactile images are prepared the more time consuming it is to compare them and select the right one. Complexity of the task is also influenced by the number of details which distinguish one image from the other. If the difference is minor and hardly noticeable, the blind student has difficulties to choose the correct option. In our opinion the third method of adaption also tests different skills than in the case of sighted students. Having drawn the picture they prove to know how to put all the computed results together and to understand how they influence the graph. On the other side, the blind student pays attention only to differences between tactile images he receives. Having computed properties of the function he/she selects that one corresponding to the results of the previous work.