When I was in high school (in the previous century!), I was taught that the value of a in the quadratic function y = ax2 determines the shape of its graph. For example, we were taught that if a > 0, the higher the value of a, the higher (and closer together) the 'arms' of the parabola would go, and conversely, the lower the value of a, the lower (or flatter) the 'arms' of the parabola would become.
However, as I only realized to my surprise some years later, this is MATHEMATICALLY WRONG! It can be shown through a series of transformations (translations, reflections, rotations & dilations (enlargements or reductions)) that any parabola can be transformed to map onto another. This proves that all parabola are SIMILAR, and that they do NOT have different shapes. Like circles, they all have the SAME SHAPE, and are merely enlargements or reductions of one another. This is shown in the 1st interactive, dynamic visual proof of the similarity of (rectangular) parabola below.
To my shock, I recently (Jan 2019) found out that, a half century later, there are still South African textbooks (and teachers) that propogate this mathematical misconception and falsehood about the effect of the value of a on the graph of the parabolic function y = ax2. As we shall see in the 2nd activity below (use the button in the sketch to navigate there), the value of a only determines the dilation (enlargement or reduction) of the parabola, i.e. how much one has 'zoomed in' or 'zoomed out', and not its shape.
Visual, dynamic proof that all parabola are similar (for a > 0, since for a < 0, only a reflection in x-axis is required)
Instructions Activity 1
a) Drag sliders p and q to translate the purple parabola with the transformation g(x) = f(x-q) + p until its vertex coincides with that of the red parabola (y = h(x) = x2).
b) Click on Show Function Plot, and then drag slider k, to enlarge/reduce the translated parabola with the dilation transformation q(x) = k.g(x/k) centred at the origin, until it coincides with the red one (y = h(x) = x2).
c) Change the parameters of the general purple parabola by dragging a (keep a > 0), b and c. Then repeat steps 1 and 2 above.
Read my 1994 article in Pythagoras at All parabola similar? Never! for more background.
Instructions Activity 2
a) Click on the 'Link to Further Illustration' Button to navigate to the 2nd activity.
b) Drag slider k to dilate (enlarge/reduce) the purple parabola y = f(x) = x2 with the shown rectangle ABCD by the dilation transformation g(x) = k.f(x/k) centred at the origin to map onto the green parabola and A'B'C'D'.
b) If g(x) = ax2 = k.f(x/k), what is the relationship between a and the scale factor k? More specifically, what changes in the value of a determine whether one is zooming in' (enlarging) or 'zooming out' (reducing)?
Further reading: Viewers might also find useful the following related High School Learning Materials from the Common Core Mathematics Curriculum for Algebra of New York State: Lesson 35.
Back to "Transformations of Graphs"
Back to "Student Explorations"
Back to "Dynamic Geometry Sketches"
Michael de Villiers, first created approx. 2004, Updated 14 May 2011, 9 January 2019.