What is the equation of y = x3
with the given transformations?
vertical stretch by a factor of 3, horizontal shift 4 units to the right, vertical shift 3 units down
The equation of the transformed function is:
y = 3(x - 4)^3 - 3
To find the equation of the given function with the specified transformations, follow these steps:
1. Start with the original equation: y = x^3.
2. Apply the vertical stretch by a factor of 3. In this case, multiply the entire equation by 3: y = 3 * x^3.
3. Apply the horizontal shift of 4 units to the right. To shift a function right, you replace x with (x - h), where h is the amount of units to shift. In this case, h = -4 (negative because it's a shift to the right). So the equation becomes: y = 3 * (x - (-4))^3.
4. Simplify the expression inside the parentheses: y = 3 * (x + 4)^3.
5. Apply the vertical shift of 3 units down. To shift a function down, you subtract the amount of units to shift from the equation. In this case, subtract 3 from the equation: y = 3 * (x + 4)^3 - 3.
Therefore, the equation of y = x^3 with the specified transformations (vertical stretch by a factor of 3, horizontal shift 4 units to the right, vertical shift 3 units down) is y = 3 * (x + 4)^3 - 3.
To find the equation of the transformed function, we need to start with the basic equation y = x^3 and apply the given transformations one by one.
1. Vertical stretch by a factor of 3:
To vertically stretch a function, we multiply the original function by the desired factor. In this case, we want to stretch the function by a factor of 3, so we multiply y = x^3 by 3 to get:
y = 3 * x^3
2. Horizontal shift 4 units to the right:
To horizontally shift a function, we replace x with (x - h), where h represents the amount of units to shift. In this case, we want to shift the function 4 units to the right, so we replace x with (x - 4) to get:
y = 3 * (x - 4)^3
3. Vertical shift 3 units down:
To vertically shift a function, we add or subtract the desired amount from the function. In this case, we want to shift the function 3 units down, so we subtract 3 from the entire function to get:
y = 3 * (x - 4)^3 - 3
So, the equation of the function with the given transformations is y = 3 * (x - 4)^3 - 3.