Shifting Graphs: Practice

Moving a graph up changes the output, so add to the function from outside. Up $5$ adds $5$ to $f(x)$.

$y = f(x) + 5$

A number inside, like $f(x - 5)$, would shift the graph sideways instead.

Moving a graph down subtracts from the output, outside the function. Down $2$ subtracts $2$ from $f(x)$.

$y = f(x) - 2$

A right shift changes the input, and right by $3$ replaces $x$ with $x - 3$ (the sign looks backward).

$y = f(x - 3)$

A left shift changes the input, and left by $4$ replaces $x$ with $x + 4$.

$y = f(x + 4)$

The $-5$ is inside the function, so it is a horizontal shift, and a minus inside moves the graph right.

$y = f(x - 5) \;\Rightarrow\; \text{right } 5$

The $+2$ is inside the function, so it is a horizontal shift, and a plus inside moves the graph left.

$y = f(x + 2) \;\Rightarrow\; \text{left } 2$

Adding $2$ outside is a vertical shift up $2$, so keep the input and raise the height by $2$.

$(3,\; 7 + 2) = (3, 9)$

Adding $4$ outside raises the whole parabola by $4$, so the vertex rises straight up.

$(0,\; 0 + 4) = (0, 4)$

In $y = (x - h)^2$ the vertex sits where the inside is $0$. Here $x - 2 = 0$ gives $x = 2$, with output $0$.

$\text{vertex} = (2, 0)$

Right $1$ replaces $x$ with $x - 1$ inside the square.

$y = (x - 1)^2$

A plus inside, $(x + 1)^2$, would shift left instead.

The $-1$ inside is a right shift by $1$, so add $1$ to the input; the height stays.

$(6 + 1,\; 5) = (7, 5)$

The $+3$ inside is a left shift by $3$, so subtract $3$ from the input; the height stays.

$(2 - 3,\; 8) = (-1, 8)$