Determine the equation for the parabola that satisfies the following conditions:

Question

Graph of y=x^2 is: (i) stretched vertically by a factor of 2, (ii) reflected about the x-axis, (iii) Translated left 4 units, (iv) Translated up 3 units.

What is the new equation?

Answer 1

(i) f(x) = 2x²

(ii) f(x) = -x²

(iii) f(x) = (x + 4)²

(iv) f(x) = x² + 3

Answer 2

To apply these transformations to the graph of y = x², we have to look at the rules for transformations.

1) Firstly, to vertically stretch the function by a factor of 2, you multiply the entire function by 2 so it becomes:

y = 2x²

2) To reflect it over the x-axis, we always multiply the function by -1:

y = (-1)(2x²)

y = -2x²

3) The rule for horizontal shift says that if we are moving left or right by a certain number of units, we add/subtract those number of units to the x in the equation. Since this transformation says to horizontally shift 4 units left, we add the x by 4 (it is usually the opposite of what you expect; add when moving left and subtract when moving right):

y = -2(x + 4)²

4) To apply a vertical shift, we add/subtract the number of units to the entire equation. Since this transformation says to vertically shift up by 3 units, we add and if it was lets say vertical shift down, then we would have subtracted 3:

y = -2(x + 4)² + 3

So, with all the transformations combined, the new equation that gets formed is y = -2(x + 4)² + 3