Transforming the Square Root Function

00:00 8 marks

Consider the function \(f(x)=\sqrt{x}\).

(a) Apply a horizontal translation of 3 units to the right and a vertical translation of 2 units upwards. Write the new function and state its new domain.

(b) Reflect the function obtained in part (a) across the x-axis, and describe how its range is altered. Provide the resulting function.

Part a

4 minutes 4 marks

Question

Apply a horizontal translation of 3 units to the right and a vertical translation of 2 units upwards to the function \(f(x)=\sqrt{x}\). Write the new function and state its new domain.

Hint Hint

Hint

Remember the translation rule: \(f(x-h)+k\) shifts the graph right by \(h\) and up by \(k\).

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Complete Answer

The translated function is \(g(x)=\sqrt{x-3}+2\) with domain \([3, \infty)\).

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Part b

4 minutes 4 marks

Question

Reflect the function obtained in part (a) across the x-axis, and describe how its range is altered. Provide the resulting function.

Hint Hint

Hint

To reflect across the x-axis, multiply the entire function \(g(x)\) by -1.

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Complete Answer

The reflected function is \(h(x)=-\left(\sqrt{x-3}+2\right)\) which simplifies to \(h(x)=-\sqrt{x-3}-2\); its range becomes \(( -\infty, -2]\).

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