Question

graph \\( = -\sqrt3{x} + 2 \\)

Explanation:

⚡ Using: combining transformations · 🆕 New: graphing radical functions
How to sketch and shift root curves

Step 1: Identify the parent function

The base function (parent function) is:

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

This is a cube root function, which has an "S-shape" (or propeller shape) passing through key points:

• \( (-8, -2) \)
• \( (-1, -1) \)
• \( (0, 0) \)
• \( (1, 1) \)
• \( (8, 2) \)

Step 2: Apply transformations

We apply two transformations to the parent function \( y = \sqrt[3]{x} \) to get \( y = -\sqrt[3]{x} + 2 \):

1. Reflection over the x-axis: The negative sign in front, \( -\sqrt[3]{x} \), flips the graph vertically.

• Positive \( y \)-values become negative, and negative \( y \)-values become positive.

1. Vertical shift up by 2 units: Adding 2 at the end, \( + 2 \), shifts the entire flipped graph upward by 2 units.

Step 3: Find key points for the new graph

We apply these transformations to our key \( x \)-values:

• For \( x = -8 \):

$$ y = -\sqrt[3]{-8} + 2 = -(-2) + 2 = 4 \implies (-8, 4) $$

• For \( x = -1 \):

$$ y = -\sqrt[3]{-1} + 2 = -(-1) + 2 = 3 \implies (-1, 3) $$

• For \( x = 0 \) (the inflection point):

$$ y = -\sqrt[3]{0} + 2 = 0 + 2 = 2 \implies (0, 2) $$

• For \( x = 1 \):

$$ y = -\sqrt[3]{1} + 2 = -1 + 2 = 1 \implies (1, 1) $$

• For \( x = 8 \):

$$ y = -\sqrt[3]{8} + 2 = -2 + 2 = 0 \implies (8, 0) $$

Step 4: Sketch the graph

1. Plot the calculated points: \( (-8, 4) \), \( (-1, 3) \), \( (0, 2) \), \( (1, 1) \), and \( (8, 0) \).
2. Draw a smooth, continuous curve through these points.
3. The curve should be nearly vertical as it passes through the inflection point at \( (0, 2) \), flattening out as it goes far to the left and right.

Answer:

To graph \( y = -\sqrt[3]{x} + 2 \):

1. Start with the standard cube root curve \( y = \sqrt[3]{x} \).
2. Flip it vertically across the x-axis (so it goes down from left to right instead of up).
3. Shift the entire curve up by 2 units.

Key points to plot:

• \( (-8, 4) \)
• \( (-1, 3) \)
• \( (0, 2) \) (the center inflection point)
• \( (1, 1) \)
• \( (8, 0) \) (the x-intercept)