Problem 35 - Algebra 2

Question

Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph. \(y=\sqrt{64 x-128}-3\)

Step-by-Step Solution

Answer

The function \(y=\sqrt{64 x-128}-3\) can be rewritten as \(y = \sqrt{64(x-2)} - 3\) for transformation-based graphing showing it has been stretched vertically by a factor of 64, shifted 2 units to the right and 3 units down. The graph starts from (2, -3) with a steep increase to the right.

Step 1: Identify the Parent Function

In the given function \(y=\sqrt{64 x-128}-3\), the parent function is \(\sqrt{x}\). This is the simplest form of the function with no transformations.

Step 2: Recognize the Changes to the Domain

The function \(y=\sqrt{64 x-128}-3\) has a domain that has been modified by the multiplier 64 and subtracted by 128 in the argument, making it \(64x-128\). Dividing by the multiplier, we have \(x-\frac{128}{64}=x-2\) rewritten inside the square root function.

Step 3: Recognize Shift and Changes to Range

The 'minus 3' outside of the square root function is a vertical downward shift by 3 units. Conversely, the 'minus 2' inside the function indicates a horizontal shift to the right by 2 units.

Step 4: Rewrite Function

With the above steps, we can rewrite the function as \(y = \sqrt{64(x-2)} - 3\). This shows the function has been stretched vertically by a factor of 64, shifted to the right by 2 units and down by 3 units.

Step 5: Describe the Graph

The graph of this function starts from the point (2, -3), and extends to the right with a vertical stretch, becoming steeper than the original square root parent function.

Key Concepts

Graphing Functions

Graphing functions is a means to visually represent the behavior of a function on a coordinate plane. It helps in understanding how the output of the function changes in response to different inputs. When graphing, the first step is often to identify the parent function. The parent function is the simplest form of the function that acts as the base to which transformations are applied. Each transformation, whether a shift, stretch, or reflection, modifies the graph in a specific and predictable way. Being able to identify these transformations can make graphing more intuitive. In this specific case, understanding the graph of the function involves following the transformation steps applied to the parent square root function.

Square Root Functions

Square root functions have the general form of \(y = \sqrt{x}\). This function has a distinct curve that starts at the origin (0,0) and extends infinitely to the right, going upwards. The graph curves because the rate of increase is not constant; each increase in \(x\) results in gradually smaller increments in \(y\). As you include transformations such as shifts or stretches, this basic shape modifies slightly. For example, multiplication inside the square root, as seen in \(64(x-2)\), can stretch or compress the graph vertically. These functions are an excellent way to explore how mathematical transformations affect dynamics of a graph.

Domain and Range

The domain of a function refers to the set of all possible input values (\(x\)-values) for which the function is defined. In the context of square root functions, the domain is typically restricted to values that make the expression under the square root non-negative. In the case \(y = \sqrt{64x-128} - 3\), the expression \(64x - 128\) must be non-negative, translating to \(x \geq 2\). This means the graph only exists starting from \(x = 2\).
The range of a function, on the other hand, refers to the possible output values (\(y\)-values). Given the vertical shift downwards by 3 units, the smallest possible \(y\)-value is -3, making the range \(y \geq -3\). Understanding the domain and range of a transformed function is crucial in accurately plotting its graph.

Horizontal and Vertical Shifts

Transformations of a function can include horizontal and vertical shifts. These shifts translate the original graph along the axes without altering its shape. A horizontal shift involves moving the graph to the left or right. For example, \(y = \sqrt{64(x-2)} - 3\) indicates a horizontal shift to the right by 2 units due to the \((x-2)\) term. This means the entire graph moves 2 units rightward along the \(x\)-axis.
Vertical shifts involve moving the graph up or down. The 'minus 3' outside the square root function in our example results in a downward shift by 3 units. Recognizing these shifts is essential when sketching graphs, as they determine the new starting point and the direction of translation on the coordinate plane.