Problem 57 - Algebra 2

Question

Rewrite each function to make it easy to graph using transformations of its parent function. Describe the graph. \(y=\sqrt{36 x+108}+4\)

Step-by-Step Solution

Answer

The rewritten function is \(y=\sqrt{36(x+\frac{3}{2})} + 4\). The graph of the function is horizontally compressed by a factor of \(36\), and shifted \(\frac{3}{2}\) units to the left and \(4\) units upwards while still maintaining the shape of a half-parabola.

Step 1: Identify the Parent Function

The parent function is the square root function \(y=\sqrt{x}\).

Step 2: Rewrite the Function

Take the given function \(y=\sqrt{36x+108}+4\), and rewrite it in this form: \(y=a \sqrt{b(x-h)} +k\), where the constants \(a\), \(b\), \(h\), and \(k\) signify transformations to the parent function. In this case, we can rewrite the function as \(y=\sqrt{36(x+\frac{3}{2})} + 4\).

Step 3: Describe the Graph

The graph is a transformation of the parent function \(y=\sqrt{x}\). The value of \(b=36\) indicates that the graph is compressed horizontally by a factor of \(36\). The term \(h=-\frac{3}{2}\) indicates a horizontal shift of \(\frac{3}{2}\) units to the left while \(k=4\) signifies a vertical shift of \(4\) units upwards. The graph still maintains the typical shape of a square root function, a half-parabola, but is narrower and shifted to the left and upwards.

Key Concepts

Parent Function

The term "parent function" refers to the simplest form of a function in a particular family of functions. For any transformations or adjustments made to a function, understanding the parent function provides a foundation. For this task, the parent function is the square root function, which is expressed as:
\[ y = \sqrt{x} \]
The graph of this parent function begins at the origin

• (0, 0) for positive values in the first quadrant only,
• rises slowly as it moves to the right.

It forms a half-parabola shape and doesn’t cross into the negative x or y regions in its simplest form. Using this basic shape, we can apply transformations to shift, stretch, or compress the graph, creating variations of this foundational shape. Understanding the parent function allows us to recognize these transformations easily.

Transformations of Functions

Transformations allow us to modify the graph of a function in various ways, creating a multitude of shapes and positions. For the function given in the exercise,
\[ y = \sqrt{36(x + \frac{3}{2})} + 4 \]
several transformations are applied to the parent square root function:

• Horizontal Compression: The coefficient of 36 inside the square root signifies a horizontal compression by a factor of \( \frac{1}{36} \). This makes the graph narrower and steeper.
  
• Horizontal Shift: The expression \((x + \frac{3}{2})\) indicates a horizontal shift 1.5 units to the left, often counterintuitive because of the positive sign.
  
• Vertical Shift: The constant \( +4 \) outside the square root signifies a vertical shift upward by four units. This means the whole graph moves up from its original position.

Transformations are powerful because they allow us to adjust functions practically without altering their basic shapes. Recognizing these transformations helps in graphing complex equations intuitively and efficiently.

Square Root Function

The square root function is a non-linear function characterized by its curved form. Its standard expression is:
\[ y = \sqrt{x} \]
The domain of the square root function is

• all non-negative real numbers
  ((0, \infty)), starting from zero and extending to positive infinity.
• The range is also all non-negative numbers, reflecting only in the first quadrant.

In a graph, this function displays a gentle, continuous curve that ascends gradually as you move along the x-axis.

• It begins at the point \((0, 0)\) (the origin),
• proceeds rightwards without touching negative axes.

It’s important to know that no real numbers satisfy the square root of a negative value. Hence, in transformations involving this function, we ensure modifications don’t lead to undefined expressions. By applying transformations, like those explored, the square root function can shift vertically, horizontally, compress, or stretch. Nevertheless, it continues to exhibit its characteristic half-parabola shape, bridging its foundational nature and modified version.