Question

graph the equation shown below by transforming the given graph of the parent function.
y = 2x³

Explanation:

Step1: Identify the parent function

The parent function for \( y = 2x^3 \) is \( y = x^3 \). The graph of \( y = x^3 \) is a cubic curve passing through the origin, symmetric about the origin, with a point at \( (1,1) \), \( (-1,-1) \), \( (2,8) \), \( (-2,-8) \) etc.

Step2: Analyze the transformation

The given function is \( y = 2x^3 \), which is a vertical stretch of the parent function \( y = x^3 \) by a factor of 2. For a vertical stretch by a factor \( a \) (where \( a>1 \)), the transformation rule is: if \( y = f(x) \) is the parent function, then \( y = a\cdot f(x) \) vertically stretches the graph by a factor of \( a \). So, for each \( x \)-value, the \( y \)-value of the parent function \( y = x^3 \) is multiplied by 2.

- For \( x = 1 \): Parent function \( y = 1^3=1 \), transformed function \( y = 2\times1 = 2 \)? Wait, no, wait the graph in the image has a point at \( (1,1) \)? Wait, no, maybe the parent function in the graph is \( y = x^3 \), and we need to transform it to \( y = 2x^3 \). Let's check the points:

- For \( x = 1 \): Parent \( y = 1^3 = 1 \), transformed \( y = 2\times1^3 = 2 \)? But the given graph has a blue dot at \( (1,1) \), maybe the parent graph is \( y = x^3 \), and we need to stretch it vertically by 2. So the new points will be:

- At \( x = -2 \): Parent \( y = (-2)^3=-8 \), transformed \( y = 2\times(-8)=-16 \)? Wait, no, the graph in the image has a blue dot at \( (-2,-8) \)? Wait, the image shows a graph with points: let's list the visible points. The green dot at origin \( (0,0) \), blue dot at \( (1,1) \), \( (2,8) \), \( (-1,-1) \), \( (-2,-8) \). Wait, that's the graph of \( y = x^3 \). So to graph \( y = 2x^3 \), we need to take each point \( (x,y) \) on \( y = x^3 \) and transform it to \( (x, 2y) \).

- For \( x = 1 \): Original \( y = 1 \), new \( y = 2\times1 = 2 \).

- For \( x = 2 \): Original \( y = 8 \), new \( y = 2\times8 = 16 \)? But the graph in the image has a blue dot at \( (2,8) \), so maybe the parent graph is \( y = x^3 \), and we need to stretch it vertically by 2. So the transformation is:

- Take the parent graph \( y = x^3 \) (with points like \( (-2,-8) \), \( (-1,-1) \), \( (0,0) \), \( (1,1) \), \( (2,8) \)) and multiply each \( y \)-coordinate by 2. So the new points will be:

- \( (-2, 2\times(-8)) = (-2, -16) \)? Wait, no, that can't be. Wait, maybe I misread the graph. Wait the given graph in the image: let's check the y-axis. The y-axis has marks from -10 to 10. The blue dot at \( (2,8) \), \( (1,1) \), \( (-1,-1) \), \( (-2,-8) \). So that's \( y = x^3 \). Now, the function to graph is \( y = 2x^3 \), so for each \( x \), \( y \) is doubled. So:

- At \( x = 1 \): \( y = 2\times1^3 = 2 \)

- At \( x = 2 \): \( y = 2\times2^3 = 16 \) (but 16 is above the y-axis limit of 10? Wait, maybe the graphing tool has a grid, and we need to plot the transformed points. So the steps to graph are:

1. Start with the parent function \( y = x^3 \), which has key points \( (-2, -8) \), \( (-1, -1) \), \( (0, 0) \), \( (1, 1) \), \( (2, 8) \).

2. Apply the vertical stretch by factor 2: multiply each y-coordinate by 2. So the new key points are:

- \( (-2, 2\times(-8)) = (-2, -16) \) (but maybe the graphing tool will show the relative stretch)

- \( (-1, 2\times(-1)) = (-1, -2) \)

- \( (0, 2\times0) = (0, 0) \)

- \( (1, 2\times1) = (1, 2) \)

- \( (2, 2\times8) = (2, 16) \)

But since the original graph has points at \( (1,1) \), \( (2,8) \), etc., to transform it, we move each point vertically: for \( (1,1) \), we go to \( (1,2) \); for \( (2,8) \), we go to \( (2,16) \) (but if the grid is up to y=10, maybe it's a stretch). The main transformation is a vertical stretch by 2, so the graph will be narrower (steeper) than the parent function \( y = x^3 \), passing through the origin, with the same symmetry (odd function, symmetric about origin), and for each x, y is twice as much as in \( y = x^3 \).

To graph it, we can plot the transformed points:

- At \( x = -1 \): instead of \( (-1, -1) \), it's \( (-1, -2) \)

- At \( x = 1 \): instead of \( (1, 1) \), it's \( (1, 2) \)

- At \( x = 2 \): instead of \( (2, 8) \), it's \( (2, 16) \) (but if the graphing tool allows, or maybe the original graph's points are scaled. The key is to recognize the vertical stretch by factor 2.

Answer:

To graph \( y = 2x^3 \) from the parent function \( y = x^3 \):

1. Identify Transformation: \( y = 2x^3 \) is a vertical stretch of \( y = x^3 \) by a factor of 2 (since \( y = 2 \cdot x^3 \), where \( a = 2 > 1 \) for \( y = a \cdot f(x) \)).
2. Transform Key Points: For the parent function’s key points \((x, x^3)\) (e.g., \((-2, -8)\), \((-1, -1)\), \((0, 0)\), \((1, 1)\), \((2, 8)\)), multiply the \( y \)-coordinate by 2:

• \((-2, -16)\), \((-1, -2)\), \((0, 0)\), \((1, 2)\), \((2, 16)\).

1. Plot and Sketch: Plot these transformed points and draw a smooth cubic curve through them, maintaining the origin symmetry (odd function) and steeper slope than \( y = x^3 \).

(Note: The graph will be vertically stretched, passing through the origin, with \( y \)-values doubled relative to \( y = x^3 \) at each \( x \).)