Problem 94 - 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. $$y=\sqrt{32-4 x}-3$$

Step-by-Step Solution

Answer

The graph is obtained by right-shifting 8, reflecting horizontally, stretching horizontally by 4, and shifting downward by 3.

Step 1: Identify the Base Function

The given function is \( y = \sqrt{32 - 4x} - 3 \). The base function here is the square root function \( y = \sqrt{x} \).

Step 2: Horizontal Transformation

The expression inside the square root, \( 32 - 4x \), suggests a horizontal transformation. To express it as \( \sqrt{x} \), consider \( y = \sqrt{-4(x - 8)} \). This indicates a horizontal shift of 8 units to the right and a reflection across the y-axis.

Step 3: Horizontal Stretch

The coefficient of \( x \) is -4, which implies a horizontal stretch by a factor of \( \frac{1}{4} \). This modifies the rate at which the function approaches its new y-values.

Step 4: Vertical Transformation

The outer \( -3 \) at the end of \( \sqrt{32 - 4x} - 3 \) implies a vertical shift. The graph is shifted downward by 3 units.

Step 5: Combine Transformations

Starting from \( y = \sqrt{x} \), apply the reflections, shifts, and stretches. Shift the base graph to the right by 8 units, reflect it across the y-axis, stretch it horizontally by 4, and finally shift it downward by 3 units.

Key Concepts

Square Root Function

The square root function is a fundamental building block in mathematics. At its core, it is expressed as \( y = \sqrt{x} \).
This function's graph is a gentle curve starting from the origin \((0, 0)\) and extending infinitely to the right along the x-axis.
It is important because it forms the basis for many transformations.

• When you input smaller values of \(x\), the resulting \( y \) value is also small, rising slowly as you move to larger \(x\) values.
• The shape of the curve is due to the square root's inherent property of converting numbers to their principal square roots.

Horizontal Shift

A horizontal shift involves moving the entire graph of a function left or right, affecting how the input variable interacts with the function. For our function, \(y = \sqrt{32 - 4x} - 3\), the term \(32 - 4x\) is key.
This illustrates a horizontal transformation.

• To see this, rewrite the inside as \(-4(x - 8)\), indicating that the graph shifts 8 units to the right because of the \( x - 8 \) term.
• This shift is opposite to what you might intuitively expect; a \(-\) inside the function moves the graph to the right.

Remember, this shift modifies where the function starts but not its shape.

Vertical Shift

Vertical shifts involve moving the graph up or down without altering its shape. In \( y = \sqrt{32 - 4x} - 3 \), the \(-3\) outside the square root applies a vertical shift.
This results in the entire graph moving downward by 3 units.

• Unlike the horizontal shift, vertical shifts change the y-values directly by adding or subtracting a constant.
• This shift affects where the function's output starts on the y-axis; here, it means that every corresponding point on the graph is lowered by 3 units.

These changes affect how the graph is positioned vertically while maintaining its overall form.

Reflections in Graphs

Reflections are transformations that flip the graph over a specific axis. In the given function, \( y = \sqrt{32 - 4x} - 3 \), a reflection occurs in the x-coefficient because of the term \(-4x\).
This results in a reflection over the y-axis.

• To see this reflection, consider \(-4(x - 8)\) as \((-4) \times (x - 8)\). The negative sign leads to the reflection.
• Reflections can change the direction of a function’s opening. Here, without the reflection, the graph of \( \sqrt{x} \) would open to the right; with it, it opens to the left.

Overall, reflections change the directionality of the graph, providing a mirrored image over the chosen axis.