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Worked solution

Complete the square before choosing transformations

Describe how the graph of y=5x2−20x+16y=5x^2-20x+16 is obtained from y=x2y=x^2 using a sequence of transformations.

CJC · Transformations · Example 10

Rewrite the quadratic

Answer

y=5(x−2)2−4y=5(x-2)^2-4.

The Teacher fills the completing-square working that is blank in the Student copy. Retain every intermediate equality:

Factor the quadratic coefficient

5x2−20x+16=5(x2−4x)+16\begin{aligned}&5x^2-20x+16\\ & =5(x^2-4x)+16\end{aligned}

Keep the constant 16 outside the bracket.

Complete the square

5(x2−4x)+16=5(x−2)2+16−5(22)=5(x−2)2−4\begin{aligned}&5(x^2-4x)+16\\ & =5(x-2)^2+16-5(2^2)\\ & =5(x-2)^2-4\end{aligned}

The added 4 inside the bracket contributes 20 after multiplication by 5.

How did this part go?

Apply the three transformations

Answer

Translate 2 units in the positive x-direction, scale parallel to the y-axis by factor 5, then translate 4 units in the negative y-direction.

The Teacher asks that the replacements be demonstrated when necessary. Use the current equation at each stage:

Translate right two

y=x2⟶y=(x−2)2y=x^2\quad\longrightarrow\quad y=(x-2)^2

Replace x by x−2x-2.

Scale vertically by five

y5=(x−2)2⟹y=5(x−2)2\frac y5=(x-2)^2\quad\Longrightarrow\quad y=5(x-2)^2

Replace y by y/5y/5.

Translate down four

y+4=5(x−2)2⟹y=5(x−2)2−4y+4=5(x-2)^2\quad\Longrightarrow\quad y=5(x-2)^2-4

Replace y by y+4y+4.

How did this part go?