Question

domain and range will always be all real numbers \\((-\infty, \infty)\\)

1. \\(y = -\sqrt3{x} + 2\\)

starting point: \\((0, 2)\\)
\a\: \\(-1\\)
domain: \\(\mathbb{r}\\)
range: \\(\mathbb{r}\\)

increase: never
decrease: \\(\mathbb{r}\\)
positive: \\((-\infty, 8)\\)
negative: \\((8, \infty)\\)

end behavior: \\(x \to -\infty, y \to \infty\\), \\(x \to \infty, y \to -\infty\\)

Explanation:

🆕 New Concept Discovered: Graphing Cube Root Functions
Analyzing key features like inflection points, intervals, and end behavior.

Step 1: Identify the Function and Key Parameters

The given function is:

$$ y = -\sqrt[3]{x} + 2 $$

This is a transformation of the parent cube root function \( y = \sqrt[3]{x} \).

• Inflection Point (or "Starting Point" for graphing purposes): The center of symmetry where the curve changes concavity. Since there is no horizontal shift inside the radical, \( x = 0 \). Substituting \( x = 0 \) gives \( y = 2 \). Thus, the inflection point is \( (0, 2) \).
• Vertical Stretch/Reflection factor "\( a \)": The coefficient in front of the radical is \( -1 \). The negative sign reflects the graph vertically across the \( x \)-axis.

Step 2: Determine Domain and Range

For any odd root function like a cube root:

• Domain: All real numbers, because you can take the cube root of any real number.

$$ (-\infty, \infty) \text{ or } \mathbb{R} $$

• Range: All real numbers, because the outputs extend from negative infinity to positive infinity.

$$ (-\infty, \infty) \text{ or } \mathbb{R} $$

Step 3: Analyze Intervals of Increase and Decrease

The parent function \( y = \sqrt[3]{x} \) is strictly increasing everywhere.
Because of the vertical reflection (\( a = -1 \)), the graph of \( y = -\sqrt[3]{x} + 2 \) is flipped upside down.

• Increase: Never
• Decrease: Over its entire domain, which is \( (-\infty, \infty) \) or \( \mathbb{R} \).

Step 4: Find Positive and Negative Intervals

To find where the function is positive (\( y > 0 \)) and negative (\( y < 0 \)), we first find the \( x \)-intercept by setting \( y = 0 \):

$$ 0 = -\sqrt[3]{x} + 2 $$

$$ \sqrt[3]{x} = 2 $$

$$ x = 2^3 = 8 $$

The graph crosses the \( x \)-axis at \( (8, 0) \).

• Since the function is strictly decreasing:
• It is above the \( x \)-axis (\( y > 0 \)) to the left of the intercept: \( (-\infty, 8) \)
• It is below the \( x \)-axis (\( y < 0 \)) to the right of the intercept: \( (8, \infty) \)

Step 5: Determine End Behavior

As \( x \) goes to positive or negative infinity, we look at the behavior of \( y = -\sqrt[3]{x} + 2 \):

• As \( x \to -\infty \), the term \( -\sqrt[3]{x} \) becomes positive and grows without bound:

$$ y \to \infty $$

• As \( x \to \infty \), the term \( -\sqrt[3]{x} \) becomes negative and decreases without bound:

$$ y \to -\infty $$

(Note: The handwritten answers on the sheet for end behavior are reversed and should be corrected.)

Answer:

• Inflection Point: \( (0, 2) \)
• "a": \( -1 \)
• Domain: \( \mathbb{R} \) or \( (-\infty, \infty) \)
• Range: \( \mathbb{R} \) or \( (-\infty, \infty) \)
• Increase: Never
• Decrease: \( \mathbb{R} \) or \( (-\infty, \infty) \)
• Positive: \( (-\infty, 8) \)
• Negative: \( (8, \infty) \)
• End Behavior:
• As \( x \to -\infty \), \( y \to \infty \)
• As \( x \to \infty \), \( y \to -\infty \)