Question

Find the function that is finally graphed after each of the following transformations is applied to the graph of \(y=\sqrt{x}\) in the order stated. (1) Reflect about the \(x\) -axis (2) Shift right 3 units (3) Shift down 2 units

Short answer

y = -√(x-3) - 2

Step-by-step solution

Reflect about the x-axis

The first transformation is a reflection about the x-axis. Reflecting the function y=√x about the x-axis changes the sign of the function. Thus, the new function becomes: y = -√x

Shift right 3 units

Next, we shift the function to the right by 3 units. To apply this transformation, we replace x with (x-3) in the function from Step 1. Thus, the new function becomes: y = -√(x-3)

Shift down 2 units

Finally, we shift the function down by 2 units. To apply this transformation, we subtract 2 from the function obtained in Step 2. Thus, the final function is: y = -√(x-3) - 2

Key concepts

Reflection

Reflecting a function about the x-axis changes the signs of the y-values of the original function. For the function \( y = \sqrt{x} \), this means that each positive y-value becomes negative. The reflection transformation is mathematically represented as multiplying the function by -1.

Therefore, if we start with \( y = \sqrt{x} \) and reflect it about the x-axis, the new function becomes:

\( y = -\sqrt{x} \).

Remember, in a reflection over the x-axis, every point \((x, y)\) on the original graph changes to \((x, -y)\). This step lays the foundation for subsequent transformations.

Horizontal Shift

A horizontal shift moves the graph of a function left or right. For a function \(y = f(x)\), shifting it to the right by 'h' units involves replacing \(x\) with \(x - h\).

From the previous step, we have the function \(y = -\sqrt{x}\).

To shift this graph right by 3 units, replace \(x\) with \(x - 3\) resulting in the transformed function:

\(y = -\sqrt{x-3}\).

It is vital to pay attention to the direction of the shift because shifting to the right involves subtracting from x, while shifting to the left involves adding to x. Think of it in terms of movement along the x-axis: a positive shift (right) contacts the values of x later than they normally would.

Vertical Shift

A vertical shift moves the graph up or down. To shift a function \(y = f(x)\) up or down by 'k' units, you add or subtract 'k' to the whole function.

Continuing from our previous function \(y = -\sqrt{x-3}\), we need to shift it down by 2 units. This involves subtracting 2 from the entire function, resulting in:

\( y = -\sqrt{x-3} - 2 \).

Vertical shifts are straightforward:

• Shifting up involves adding a positive number.
• Shifting down involves subtracting a number.

This final step shifts the entire graph of the function downward, adjusting every y-value accordingly. The crucial point is that vertical shifts impact the function's output directly, while horizontal shifts modify the input.