Functions: Question 10

Worked solution

Part (a): $y = f(x) - 5$

Subtracting a constant outside the function translates the graph vertically down, leaving the $x$-coordinate unchanged. Here the shift is $5$ down:

$(6,\ -2-5) = (6, -7)$

Part (b): $y = f(x+3)$

Adding a constant inside the function translates the graph horizontally: $y=f(x+c)$ shifts every point $c$ units to the left (for $c>0$), leaving the $y$-coordinate unchanged. Here $c=3$:

$(6-3,\ -2) = (3, -2)$

Part (c): $y = \frac{1}{2}f(x)$

Multiplying the output of $f$ by a constant is a vertical stretch, scale factor $\tfrac{1}{2}$: the $x$-coordinate is unchanged, and the $y$-coordinate is scaled:

$\left(6,\ -2 \times \tfrac{1}{2}\right) = (6, -1)$

Part (d): $y = 4f(x+3) - 1$

This combines a horizontal translation with a vertical stretch and a vertical translation. Work through the transformations in the order they’re applied to a point on the original curve.

Step 1, horizontal shift: from part (b), the point $(6,-2)$ on $y=f(x)$ corresponds to $(3,-2)$ on $y=f(x+3)$, since $f(3+3)=f(6)=-2$.

Step 2. Vertical stretch: multiply the $y$-value by $4$: $4 \times (-2) = -8$

Step 3. Vertical shift: subtract $1$: $-8 - 1 = -9$

So the point on $y=4f(x+3)-1$ is: $(3, -9)$

Final answers

• (a) $(6, -7)$
• (b) $(3, -2)$
• (c) $(6, -1)$
• (d) $\boxed{(3, -9)}$