Question

graph \\(y = 2\sqrt3{x+5} - 4\\)

Explanation:

⚡ Using: combining transformations · 🆕 New: Graphing Cube Root Functions
S-shaped curves shifted and stretched

Step 1: Identify the parent function and key point

The parent function is:

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

Its reference point (inflection point) is at \( (0,0) \).

Step 2: Apply transformations

For the function \( y = 2\sqrt[3]{x+5} - 4 \):

• Horizontal shift: \( x + 5 \) shifts the graph left by \( 5 \) units.
• Vertical stretch: The multiplier \( 2 \) stretches the graph vertically by a factor of \( 2 \).
• Vertical shift: The \( -4 \) shifts the graph down by \( 4 \) units.

The new inflection point moves from \( (0,0) \) to:

$$ (-5, -4) $$

Step 3: Calculate key points for graphing

Choose \( x \)-values around the inflection point \( x = -5 \) that yield perfect cubes under the radical:

1. At \( x = -5 \):

$$ y = 2\sqrt[3]{-5+5} - 4 = 2(0) - 4 = -4 \implies (-5, -4) $$

1. At \( x = -4 \):

$$ y = 2\sqrt[3]{-4+5} - 4 = 2(1) - 4 = -2 \implies (-4, -2) $$

1. At \( x = -6 \):

$$ y = 2\sqrt[3]{-6+5} - 4 = 2(-1) - 4 = -6 \implies (-6, -6) $$

1. At \( x = 3 \):

$$ y = 2\sqrt[3]{3+5} - 4 = 2(2) - 4 = 0 \implies (3, 0) $$

1. At \( x = -13 \):

$$ y = 2\sqrt[3]{-13+5} - 4 = 2(-2) - 4 = -8 \implies (-13, -8) $$

Step 4: Sketch the graph

1. Plot the center inflection point at \( (-5, -4) \).
2. Plot the surrounding points: \( (-6, -6) \), \( (-4, -2) \), \( (3, 0) \), and \( (-13, -8) \).
3. Draw a smooth, continuous S-shaped curve passing through these points, flattening out horizontally as it goes further left and right.

Answer:

To graph \( y = 2\sqrt[3]{x+5} - 4 \), plot the following key points and connect them with a smooth S-shaped curve:

• Inflection Point: \( (-5, -4) \)
• Additional Points:
• \( (-6, -6) \)
• \( (-4, -2) \)
• \( (3, 0) \) (the x-intercept)
• \( (-13, -8) \)