A parent function is the simplest function in its family (Kuang & Gilman, 2011). It is from this parent function that other functions are ‘born’ from translations, reflections and other transformations. Sometimes, the parent function is referred to as the basic function. In this study, the researcher concentrated only on the vertical and horizontal translations. In exponential functions, the parent function is expressed as 𝑓(𝑥) = 𝑏𝑥, where b is a positive rational (b > 0; b≠ 1). With regard to the hyperbola, the parent function is expressed as 𝑔(𝑥) = 𝑎
𝑥. Twelve participants (50%) described the parent function as either a standard or original function, but none of them used the term “parent function”. A standard quadratic function is
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written in the form of 𝑓(𝑥) = 𝑎(𝑥 − 𝑝)2+ 𝑞, while its parent function is 𝑓(𝑥) = 𝑥2. In the next few paragraphs, the researcher describes how participants discussed the parent function. Participant R and Participant W used the term “basic graph” to describe the parent function. The use of the term “basic graph” indicates that the learners were able to describe the parent function. These two participants were presented with a graph in the form of an algebraic representation of a function. They could have referred to the graph as an equation. They named the algebraic representation of an exponential function as follows:
Participant W: So, f is a exponential and y is a basic graph but f has obtain a shift in y 1unit to the left and then g(x) is a hyperbola graph and y is a basic form of a hyperbola but g(x) is obtain by shifting y 3 units left and 1 unit upwards.
Interviewer: What do you think? Participant R: It is the same thing.
While Participant W posits function f as an exponential function, she did not state for which function was it the basic graph. She only explained the horizontal shift of the function and said nothing of the horizontal for both functions. She just agreed with Participant W, and they both did not provide more information relating to their mathematical discourse. However, both participants used mathematical words. In the meantime, Participant Z used both the standard and original functions as though they were synonymous. She named the functions as exponential function and the hyperbola. She went on to describe the relationship between the parent function and the transformed function. She described the relationship between the algebraic representations 𝑦 = 3𝑥 and 𝑓(𝑥) = 3𝑥+1− 9 as follows:
Participant Z: The question is given as okay. 𝑦2, 𝑦 = 3𝑥, is the standard form. And f (x) which is y to the, which is three to the power x plus one minus nine (𝑓(𝑥) = 3𝑥+1− 9) . Here it will do a shifting one: a shifting or a translation both vertical and horizontal translation
Interviewer: What do you mean by standard form.
Participant Z: Standard form means it is not shifted. Just original. , it not shifted. where x , there is no translation
Participant Z’s response above suggests that she was able to described the relationship between the parent function and the transformed function as a translation. Therefore, her use of the word “translation” was mathematical. It was unlike some participants who did not use the term “translation” at all. She also used the term “standard form”. When asked for the meaning of “standard form”, she stated that it“ means it is not shifted, just original”. What she meant was that 𝑦 = 3𝑥, was not translated, but 𝑓(𝑥) = 3𝑥+1− 9 has been transformed. Language use might have affected how Participant Z expressed herself, but her intentions were clear. For her, a standard form was the parent function. She did not use words
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appropriately, as indicated by her use of “original” and “standard form” as though they were synonymous. A standard function is not the same as the parent function, nor is it known as an original function.
Participant AA, who was a partner of Participant Z during the interviews, spoke of translations as Participant OO spoke of eliminations. However, Participant AA used the “translations” in his response. The question was asked: Explain how f (𝑥)= 𝟑𝒙+𝟏− 𝟗 relates to 𝑦 = 3𝑥 and g(𝒙) = 𝟐
𝒙+𝟑+ 𝟏 relates to = 2
𝑥 , to which he responded as follows:
Participant AA: So now ,what we did here is, we saw that 𝑦 = 3𝑥 we saw that the -9 and the +1 were Participant OO eliminated from the whole equation so we thought, see it here, y to the power 3𝑥 is the standard form of an exponential function and f is translated graph of y = 3𝑥 is obtained by shifting y 9 units down and 1 unit to the left.
Interviewer: Ok, go on.
Participant AA: And the hyperbola 𝑦 =𝑥2 is the standard form of a hyperbola and g is obtain by shifting the standard graph 3 units to the left and 1unit up.
To some extent, Participant AA explained the relationship from the transformed function to the parent function, yet all functions were transformed from the parent function. However, he self-corrected and used a proper mathematical term, “translation”. He explained correctly that f was a translation of the parent function, and went on to explain the hyperbola accordingly.
Although Participant AA could explain the relationship between the parent function and its offspring, he still used “standard form” instead of “parent function”. His use of words was somewhere between “colloquial” and “mathematical”. At the same time, Participant AA’s interpretation of the symbolic visual mediator was classified as “construed”, since he was able to show the relationship between the two functions, as well as to explain the translation. There was evidence of a developing mathematical discourse in the functions discourse because Participant AA could talk of the relationship between the parent function and its offspring. Participant BB and Participant CC both used the term “original function” instead of “parent function”. These two participants did not show or state how the two functions were related. Following is the trajectory of the interview with the two participants.
Participant BB: So f (𝑥)= 3𝑥+1− 9. It is an exponential function and it is related to y = 3𝑥 because y = 3𝑥. 𝐼t is the original graph of the function above and also 𝑔(𝑥) = 2
𝑥+3+ 1 is a hyperbolic function and it is related to 𝑦 =2
𝑥 since 𝑦 = 2
𝑥 it an original function. Interviewer: What do you think?
Participant CC: Yes. f(x), f is an exponential graph because it goes y = 3𝑥 I can see it is the original graph of this graph
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Participant CC and Participant BB agreed on the use of the term “original graph”. From their statements, it was clear that they meant the parent graph. The representation that they were referring to, is the algebraic representation of a function. The two participants did not explain how they related the two functions. The researcher classified their use of words as “colloquial”, because they referred to the parent function as an original graph, and to the algebraic representation of the function as a graph. They both interpreted the two pairs of symbolic representations and visual mediator mathematically, and the researcher classified this interpretation as “construed”. In this context, these two participants exhibit a growing mathematical discourse.
Participant X and Participant DD also spoke of the relationship between y = 3𝑥 and f (𝑥)=