Translating a graph

To shift by $p$ in the $x$-direction, replace $x$ with $x-p$. To shift by $q$ in the $y$-direction, add $q$ to the right-hand side.

For example, $y=(x+2)^2-2$ reads as $(x-(-2))^2-2$, so it is shifted by $-2$ in the $x$-direction and $-2$ in the $y$-direction, with vertex $(-2,\,-2)$.

A translation does not change the shape, so the coefficient of $x^2$ stays 1. A graph like $y=4x^2$, whose coefficient has changed, cannot be obtained by translating $y=x^2$.

Examples

1. 01What is the x-coordinate of the vertex?$\displaystyle y=(x-1)^2-2$AnswerHide

   Answer$\displaystyle 1$

   Steps

   1. $\displaystyle x-1 = 0$$\displaystyle \Rightarrow x = 1$
2. 02Which one cannot be obtained by translating $y=x^2$?$\displaystyle$A$\displaystyle y=(x-4)^2+5$B$\displaystyle y=(x-4)^2-1$C$\displaystyle y=(x+4)^2-3$D$\displaystyle y=4x^2$AnswerHide

   Answer$\displaystyle y=4x^2$
3. 03Shift by -3 in the y-direction. What is the equation?$\displaystyle y=x^2$AnswerHide

   Answer$\displaystyle y=x^2-3$
4. 04$y=x^2$ shifted by $-1$ in the $x$-direction. What goes in □?$\displaystyle y=(x \;\square\; 1)^2$AnswerHide

   Answer$\displaystyle +$
5. 05What is the x-coordinate of the vertex?$\displaystyle y=(x-3)^2+1$AnswerHide

   Answer$\displaystyle 3$

   Steps

   1. $\displaystyle x-3 = 0$$\displaystyle \Rightarrow x = 3$
6. 06What is the x-coordinate of the vertex?$\displaystyle y=(x+1)^2+4$AnswerHide

   Answer$\displaystyle -1$

   Steps

   1. $\displaystyle x+1 = 0$$\displaystyle \Rightarrow x = -1$

FAQ

Q1Shifting right by 3 gives $x-3$. Why a minus sign?AnswerHide

A

On the shifted graph, the height at $x=3$ must equal the original height at $x=0$. Putting $x=3$ into $x-3$ gives $0$, so the original vertex moves to $x=3$.

Q2Is $y=2x^2+1$ a translation of $y=x^2$?AnswerHide

A

No. The coefficient of $x^2$ is 2, so the shape (how wide it opens) has changed. It is $y=2x^2$ shifted by 1 in the $y$-direction.