Question

graph the following function by moving the green and blue dots (if necessary).
y = 4x³ - 4

Explanation:

Step1: Identify Key Points

For the function \( y = 4x^3 - 4 \), we can find some key points by plugging in \( x \)-values.

• When \( x = 0 \): \( y = 4(0)^3 - 4 = -4 \)? Wait, no, wait the green dot is at (0,0) in the graph, but let's recalculate. Wait, maybe the graphing tool has a different setup. Wait, actually, let's find the correct points.
• When \( x = 1 \): \( y = 4(1)^3 - 4 = 4 - 4 = 0 \).
• When \( x = -1 \): \( y = 4(-1)^3 - 4 = -4 - 4 = -8 \)? Wait, no, the blue dot on the left is at \( x=-1 \), let's check. Wait, maybe the graph is a cubic function. The parent function \( y = x^3 \) has a point at (0,0), (1,1), (-1,-1). For \( y = 4x^3 - 4 \), let's find the y-intercept: when \( x = 0 \), \( y = -4 \). Wait, the green dot in the graph is at (0,0), which might be a mistake, but maybe the graphing tool is for a different function? Wait, no, the function is \( y = 4x^3 - 4 \). Let's find the correct points:
• \( x = 0 \): \( y = -4 \)
• \( x = 1 \): \( y = 4(1) - 4 = 0 \)
• \( x = -1 \): \( y = 4(-1) - 4 = -8 \)
• \( x = 2 \): \( y = 4(8) - 4 = 28 \) (but that's too big for the graph)

Wait, maybe the graph in the image is incorrect, but the process is to plot the function \( y = 4x^3 - 4 \) by finding points. The key is to recognize the cubic function's shape (S - shaped, passing through (0, -4), (1, 0), (-1, -8)). But since the graphing tool has dots, we need to move them to the correct positions. The green dot (vertex - like for cubic? No, cubic has a point of inflection at (0, -4) for this function). Wait, maybe the initial graph is wrong, but the correct way is to plot the function by calculating points.

Step2: Adjust Dots

• The green dot (maybe the inflection point) should be at \( (0, -4) \), but in the graph, it's at (0,0). So we need to move the green dot down to (0, -4).
• The blue dot at \( x = 1 \): when \( x = 1 \), \( y = 0 \), so the blue dot at \( x = 1 \) should be at (1, 0).
• The blue dot at \( x = -1 \): when \( x = -1 \), \( y = 4(-1)^3 - 4 = -8 \), so it should be at (-1, -8).
• The other blue dot (at \( x = 8 \)? No, maybe \( x = 2 \), but \( y = 28 \) is too big. Wait, maybe the graphing tool is simplified. Alternatively, the function \( y = 4x^3 - 4 \) is a cubic function with a vertical stretch by 4 and a vertical shift down by 4. So the shape is similar to \( y = x^3 \), but steeper and shifted down.

Answer:

To graph \( y = 4x^3 - 4 \), move the green dot (inflection point) to \( (0, -4) \), the blue dot at \( x = 1 \) to \( (1, 0) \), the blue dot at \( x = -1 \) to \( (-1, -8) \), and adjust other dots to follow the cubic curve (steeper than \( y = x^3 \), shifted down 4 units).