Graph Translation: Equation of y = f(x) Moved Left and Down

Grades 9-10

Question: The graph of y = f(x) is translated left by 3 units and down by 2 units. Write down the equation of the new graph

In the HKDSE Compulsory Part syllabus, the transformation of graphs is a key topic. When a graph

y = f(x)

undergoes a translation, we modify the equation based on whether the movement is horizontal or vertical.

Horizontal Translation (Left/Right): This transformation affects the input of the function ( $x$ ).
- To translate the graph left by $h$ units, we replace $x$ with $(x + h)$ .
- To translate the graph right by $h$ units, we replace $x$ with $(x - h)$ .
- Reasoning: If we move the graph 3 units to the left, a point that originally occurred at $x=0$ now occurs at $x=-3$ . For the function to "see" the same value it used to see at $0$ , we must add 3 to the new $x$ coordinate.
Vertical Translation (Up/Down): This transformation affects the output of the function ( $y$ ).
- To translate the graph up by $k$ units, we add $k$ to the entire function: $y = f(x) + k$ .
- To translate the graph down by $k$ units, we subtract $k$ from the entire function: $y = f(x) - k$ .

Let the original equation be

y = f(x)

Step 1: Translate left by 3 units Replace

x

with

(x + 3)

. The intermediate equation is:

\begin{align*} y = f(x + 3) \end{align*}

Step 2: Translate down by 2 units Subtract 2 from the right-hand side of the equation. The final equation is:

\begin{align*} y = f(x + 3) - 2 \end{align*}

Final Answer: The equation of the new graph is

y = f(x + 3) - 2

In the diagram below:

The blue curve represents the original graph $y = f(x)$ .
The grey dashed curve shows the intermediate step after shifting 3 units to the left.
The red curve represents the final transformed graph $y = f(x + 3) - 2$ .
Notice how the vertex (or any reference point) moves from $(0, 0)$ to $(-3, 0)$ and finally to $(-3, -2)$ .

法(1) [Transformation of graphs]:

y = f(x) \xrightarrow{\text{left 3}} y = f(x + 3) \xrightarrow{\text{down 2}} y = f(x + 3) - 2

(2 marks)

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