• $\ f(x - a)$ shifts the graph right by $\ a$ units.

• $\ f(x + a)$ shifts the graph left by $\ a$ units.

Example: Given $\ f(x) = x^2 :$

• $\ f(x - 2) = (x - 2)^2$ shifts the graph of $\ x^2$ 2 units to the right.
• $\ f(x + 3) = (x + 3)^2$ shifts the graph of $\ x^2$ 3 units to the left.

• $\ f(x) + b$ shifts the graph up by $\ b$ units.

• $\ f(x) - b$ shifts the graph down by $\ b$ units.

Example: Given $\ f(x) = x^2$:

• $\ f(x) + 4 = x^2 + 4$ shifts the graph of $\ x^2$ 4 units up.
• $\ f(x) - 5 = x^2 - 5$ shifts the graph of $\ x^2$ 5 units down.

Example: Given $\ f(x) = \sin(x)$:

• $g(x) = \sin(x - \frac{\pi}{2}) + 3$ shifts the graph of $\sin(x) \ \frac{\pi}{2}$ units to the right and 3 units up.

• Horizontal Translations: Imagine moving the entire graph left or right along the $x$-axis. The $y$-values of points on the graph remain the same, but the $x$-values change.

• Vertical Translations: Imagine shifting the entire graph up or down along the $y$-axis. The $x$-values of points on the graph remain the same, but the $y$-values change.

Example 1: Graph Translation

The graph of $y=f(x)y = f(x)y=f(x)$ is translated by the vector $(3−2)\begin{pmatrix} 3 \\ -2 \end{pmatrix}(3−2)$. Write down the equation of the translated graph.

Step-by-Step Solution:

1. Understand the translation vector:

• The vector ($3−2$) indicates a translation 3 units to the right (positive $x$-direction) and 2 units down (negative $y$-direction). $(3−2)\begin{pmatrix} 3 \\ -2 \end{pmatrix}$

2. Translate the graph:

• To translate a graph by a units horizontally and b units vertically, the equation $y=f(x)$ becomes: $y=f(x−a)+b$ $aa$

$bb$

$y=f(x)y = f(x)$

$y=f(x−a)+by = f(x - a) + b$

• Here, $a=3$ and $b=−2$, so the translated graph equation is: $y=f(x−3)−2$ $a=3a = 3$

$b=−2b = -2$

$y=f(x−3)−2y = f(x - 3) - 2$

Final Answer: The equation of the translated graph is $y=f(x−3)−2y = f(x - 3) - 2y=f(x−3)−2.$

Example 2: Translation of a Shape

A triangle with vertices at $A(1,2)A(1, 2)A(1,2) \ , B(3,4)B(3, 4)B(3,4)$, and $C(5,1)C(5, 1)C(5,1)$ is translated by the vector $(−23)\begin{pmatrix} -2 \\ 3 \end{pmatrix}(−23)$. Find the coordinates of the new vertices.

Step-by-Step Solution:

3. Understand the translation vector:

• The vector ($−23$) indicates a translation 2 units to the left (negative $x$-direction) and 3 units up (positive $y$-direction). $(−23)\begin{pmatrix} -2 \\ 3 \end{pmatrix}$

4. Translate each vertex:

• For $A(1,2)$: $A′=(1−22+3)=(−1,5)$ $A(1,2)A(1, 2)$

$A′=(1−22+3)=(−1,5)A' = \begin{pmatrix} 1 - 2 \\ 2 + 3 \end{pmatrix} = (-1, 5)$

• For $B(3,4)$: $B′=(3−24+3)=(1,7)$

$B(3,4)B(3, 4)$

$B′=(3−24+3)=(1,7)B' = \begin{pmatrix} 3 - 2 \\ 4 + 3 \end{pmatrix} = (1, 7)$

• For $C(5,1)$: $C′=(5−21+3)=(3,4)$

$C(5,1)C(5, 1)$

$C′=(5−21+3)=(3,4)C' = \begin{pmatrix} 5 - 2 \\ 1 + 3 \end{pmatrix} = (3, 4)$

Final Answer: The coordinates of the translated vertices are:

$A′(−1,5)A'(-1, 5)A′(−1,5)$ , $B′(1,7)B'(1, 7)B′(1,7) \ ,$ and $C′(3,4)C'(3, 4)C′(3,4).$

Example 3: Translation of a Function

The function $y=2x2+3xy = 2x^2 + 3xy=2x2+3x$ is translated 4 units to the left and 5 units up. Write down the equation of the new function.

Step-by-Step Solution:

5. Translate horizontally:

• A translation of 4 units to the left affects the $x$-variable, changing $x$ to $x+4$. $xx$

$xx$

• The equation becomes:

6. Translate vertically:

• A translation of 5 units up adds 5 to the entire function:

Final Answer: The equation of the translated function is:

$y=2(x+4)2+3(x+4)+5y = 2(x + 4)^2 + 3(x + 4) + 5y=2(x+4)2+3(x+4)+5.$

Exam Tip:

When asked to describe or apply translations in an exam, always clearly state the direction and magnitude of the shift. Sketching a graph can help visualize and confirm the correct transformation.