Problem 93 - A Graphical Approach to Precalculus with Limits

Question

Use transformations to explain how the graph of the given function can be obtained from the graphs of the square root function or the cube root function. $$y=\sqrt{4 x+16}+4$$

Step-by-Step Solution

Answer

Shift left by 4, compress horizontally by 1/2, then shift up by 4.

Step 1: Identify the Base Function

The base function here is the square root function, which is represented as \( y = \sqrt{x} \). This will be our starting point for transformations.

Step 2: Simplify Inside the Function

We need to express \( \sqrt{4x+16} \) in a simpler form. Factor the expression inside the square root: \( 4x + 16 = 4(x + 4) \). Thus, the function can be rewritten as \( y = \sqrt{4(x+4)} + 4 \).

Step 3: Apply Horizontal Compression

Since there is a multiplier 4 inside the square root, we apply a horizontal compression by a factor of 1/2. This is because the general form \( \sqrt{c(x-h)} \) compresses the graph horizontally by \( 1/\sqrt{c} \). Our new base function is \( y = \sqrt{x} \), compressed to \( y = \sqrt{4x} \) which affects the graph by compressing it horizontally.

Step 4: Apply Horizontal Translation

The term \( \sqrt{4(x+4)} \) can be interpreted as a horizontal shift. The \( x+4 \) inside the square root shows a horizontal translation to the left by 4 units. Thus, the base function \( \sqrt{x} \) shifts to \( \sqrt{x+4} \).

Step 5: Apply Vertical Translation

Finally, the +4 outside the square root indicates a vertical shift. Every point on the graph is moved up 4 units due to \( +4 \) in \( \sqrt{4(x+4)} + 4 \).

Key Concepts

square root function

Understanding the square root function is the foundation for mastering function transformations. The square root function is generally represented as \( y = \sqrt{x} \). This function features a smooth curve that begins at the origin, (0,0), and only exists in the first quadrant of the coordinate plane.

• It's crucial to remember that \( \sqrt{x} \) is only defined for non-negative values of \( x \), as square roots of negative numbers are not real.
• The graph of this function rises steadily but slows down as \( x \) increases.

We use the square root function as a base when applying transformations because it helps us understand how the graph behaves under different modifications, which is the main focus of this exercise.

horizontal compression

Horizontal compression is an essential transformation that acts on the \( x \)-coordinates of the function. When dealing with a function like \( \sqrt{c(x)} \), if \( c \) is greater than 1, the graph compresses horizontally.

• This compression means that the graph will grow steeper and closer to the y-axis.
• The factor by which the graph compresses is \( \frac{1}{\sqrt{c}} \).

In our specific exercise with \( \sqrt{4(x+4)} \), a compression occurs due to the factor of 4 inside the square root. This compresses the graph horizontally by a factor of \( \frac{1}{\sqrt{4}} = \frac{1}{2} \). Thus, every point on the graph of \( y = \sqrt{x} \) gets moved toward the y-axis, making the rise of the graph steeper than its original.

horizontal translation

Horizontal translations move a graph left or right. For the function \( \sqrt{c(x-h)} \), the \( h \) value dictates this shift. When \( h \) is positive, the function shifts to the right, and when negative, it moves to the left.

• In the function \( \sqrt{4(x+4)} \), we see \( x+4 \) inside the square root.
• This indicates a shift to the left by 4 units.

The graph moves every point from the base graph \( y = \sqrt{x} \) to the left by 4 units, changing the starting point of the graph on the x-axis from (0,0) to (-4,0). Understanding this movement helps us see where the curve originates after transformations.

vertical translation

Vertical translation is another type of function transformation, affecting the y-coordinates. This movement is indicated by any constant added or subtracted from the function.

• For instance, in our function \( \sqrt{4(x+4)} + 4 \), the +4 shifts the entire graph upward by 4 units.
• Every point on the transformed graph is elevated, preserving the graph's shape.

This upward shift alters the rising path of the square root function so that it starts at a higher point than it would on the base graph. Adding this transformation solidifies how we perceive the positioning of graphs in a coordinate plane, simplifying complex graphs into understandable forms.