How do you write the equation for translating $y=f(x) 4$ units left and 3 units down?

Conditions

• The starting graph has equation $y=f(x)$.
• The intended transformation is a horizontal translation 4 units left and a vertical translation 3 units down.

Reasoning, step by step

1. Start with the base equation $y=f(x)$.
2. Apply the horizontal translation 4 units left by inserting $+4$ inside the function argument, yielding $y=f(x+4)$.
3. Apply the vertical translation 3 units down by subtracting 3 outside the function, yielding $y=f(x+4)-3$.

Example

The board shows the progression from $y=f(x)$ to $y=f(x+4)-3$, with the speaker stating that moving left uses plus inside the parentheses and three down is just minus three.

Common misconceptions

• Confusing a leftward shift with a minus sign; a left translation is represented by adding inside the argument, as in $f(x+4)$.
• Applying the vertical shift inside the function argument instead of outside.

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Explore next

• Why does moving left use plus inside the function argument?
• How do you write the equation for translating 5 units right and reflecting across the x-axis?