What is the equation of y=x^3 with the given transformations? Please explain your answer
a vertical stretch by a factor of 3; a reflection in the x-axis; a vertical translation 2 units down; and a horizontal translation 2 units left
To find the equation of the graph after the given transformations, we need to apply each transformation one by one.
Starting with the equation y = x^3, let's go through each transformation:
1. Vertical stretch by a factor of 3:
To vertically stretch the graph by a factor of 3, we can multiply the y-value of each point on the graph by 3. This is done by multiplying the equation by 3: y = 3 * x^3.
2. Reflection in the x-axis:
To reflect the graph in the x-axis, we change the sign of the y-values in the equation. Therefore, the equation becomes y = -3 * x^3.
3. Vertical translation 2 units down:
To vertically translate the graph 2 units down, we subtract 2 from the y-value of each point. This changes the equation to y = -3 * x^3 - 2.
4. Horizontal translation 2 units left:
To horizontally translate the graph 2 units left, we subtract 2 from the x-value. Therefore, the final equation is y = -3 * (x + 2)^3 - 2.
In summary, after the given transformations, the equation of y = x^3 becomes y = -3 * (x + 2)^3 - 2.