How to apply multiple transformations to y=x^2 in the correct order

Expert intro: With chained transformations, the order is the whole game. Horizontal changes act on the input, vertical changes act on the output.

Detailed walkthrough

Start from the base function

$f(x)=x^2.$

The intended transformations are:

• vertical reflection and dilation by factor 4
• horizontal dilation by factor 3
• shift left 5
• shift up 12

Student B: translate, then dilate

If we shift left 5 first, we get

$(x+5)^2.$

Then horizontal dilation by 3 gives

$(3x+5)^2 \text{ or equivalently } (3(x+5))^2$

depending on the exact coordinate rule being used in the course context. After the vertical changes:

$g(x)=-4(3(x+5))^2+12.$

Student A: dilate, then translate

If we dilate first,

$g(x)=-4x^2,$

then apply the horizontal change and shift. The key point is that the final algebraic form is not generally the same if the order of horizontal and translation steps changes.

Comparison

• Vertical shifts are applied to the output.
• Horizontal shifts and stretches are applied to the input.
• The order can change the final expression when more than one horizontal change is involved.

For the clean standard form, be careful to rewrite each step with the correct input substitution before expanding.

💡 Pitfall guide

Students often mix up a horizontal stretch with a vertical stretch. Another common issue is writing $(x-5)^2$ for a left shift; a left shift by 5 should usually appear as $(x+5)^2$ in function notation.

🔄 Real-world variant

If the vertical reflection were removed, the same chain would become

$g(x)=4(3(x+5))^2+12$

in the same setup. If the stretch factor were 2 instead of 3, every input term would change accordingly, but the method would stay the same.

🔍 Related terms

parabola transformation, horizontal dilation, function notation