Problem 71 - Precalculus Essentials

Question

Begin by graphing the square root function, \(f(x)=\sqrt{x} .\) Then use transformations of this graph to graph the given function. $$h(x)=-\sqrt{x+2}$$

Step-by-Step Solution

Answer

Starting with the graph of \(f(x)=\sqrt{x}\), the graph of \(h(x)=-\sqrt{x+2}\) can be obtained by shifting the graph of the base function 2 units to the left and then reflecting it about the x-axis.

Step 1: Graph the Base Function

Start by graphing the base function \(f(x)=\sqrt{x}\). This function is a half parabola that opens upwards, starting from the origin (0,0), and it only exists for x values greater than or equal to 0.

Step 2: Apply the Horizontal Shift

The \(x+2\) inside the square root in \(h(x)=-\sqrt{x+2}\) indicates a shift to the left by 2 units. So, every point on the graph of \(f(x)\) should be shifted 2 units to the left to transform it into \(h(x)\).

Step 3: Apply the Vertical Reflection

The negative sign before the square root in \(h(x)=-\sqrt{x+2}\) indicates a vertical reflection. So, every point on the shifted graph should be reflected about the x-axis. This means if a point is at (a, b), it will now be at (a, -b).

Key Concepts

Base Function Graph

Understanding the graph of the base function, \(f(x)=\sqrt{x}\), is crucial when dealing with square root functions. This base function represents a half parabola that opens upwards, with its vertex located at the origin \( (0,0) \). By default, it only defines values for \(x \geq 0\) since square roots of negative numbers are not real numbers.

Visualize the graph as a smooth curve starting at the origin and gradually increasing as \(x\) gets larger. Remember, the rate at which the curve rises decreases as \(x\) increases; it's steep at the start and flattens out. This distinctive shape is important for recognizing how transformations will affect the graph.

Horizontal Shift Transformations

To grasp horizontal shift transformations, imagine sliding the entire graph of a function left or right along the \(x\)-axis. The horizontal shift is determined by the addition or subtraction within the argument of the function.

When dealing with \(h(x)=-\sqrt{x+2}\), the \(x+2\) inside the square root suggests a horizontal shift. Since we are adding 2, contrary to our intuition, this translates to moving each point of the base function graph \(f(x)\) left by 2 units.

Visualizing the Shift

It's helpful to pick a few key points on the original graph and shift those points to the left by the same amount. This way, you create a new set of points that map out the newly transformed function on the graph.

Vertical Reflection Transformations

Vertical reflection transformations flip the graph of the function over the \(x\)-axis. Detecting a vertical reflection is straightforward: look for a negative sign before the main function expression.

In the case of \(h(x)=-\sqrt{x+2}\), the minus sign before the square root indicates this vertical reflection. Each point on the previously shifted graph will now be mirrored across the \(x\)-axis. If a point on the shifted graph is at \( (a, b) \), after the vertical reflection, it will be placed at \( (a, -b) \).

Effect on the Graph

This mirroring transformation flips the 'half parabola' of the square root graph upside down. As a result, for our example, the once upwards opening curve now opens downwards. By combining this understanding with the previous shift, the final graph of \(h(x)\) takes shape distinctly different from our starting base function \(f(x)\).