1. **State the problem:** We start with the parabola equation $y = x^2$ and apply transformations to find the new equation in the form $y = (x - h)^2 + k$ where $h$ and $k$ represent horizontal and vertical shifts respectively. 2. **Formula and rules:** The general form after shifting is: $$y = (x - h)^2 + k$$ - Moving right by $a$ units means $h = a$ (positive). - Moving left by $a$ units means $h = -a$ (negative). - Moving up by $b$ units means $k = b$ (positive). - Moving down by $b$ units means $k = -b$ (negative). 3. **Apply transformations:** **a)** Moves 3 units right: $$h = 3, k = 0$$ Equation: $$y = (x - 3)^2 + 0 = (x - 3)^2$$ **b)** Moves 4 units down: $$h = 0, k = -4$$ Equation: $$y = (x - 0)^2 - 4 = x^2 - 4$$ **c)** Moves 2 units left: $$h = -2, k = 0$$ Equation: $$y = (x - (-2))^2 + 0 = (x + 2)^2$$ **d)** Moves 5 units up: $$h = 0, k = 5$$ Equation: $$y = (x - 0)^2 + 5 = x^2 + 5$$ **e)** Moves 7 units down and 6 units left: $$h = -6, k = -7$$ Equation: $$y = (x - (-6))^2 - 7 = (x + 6)^2 - 7$$ **f)** Moves 2 units right and 5 units up: $$h = 2, k = 5$$ Equation: $$y = (x - 2)^2 + 5$$ 4. **Summary:** - a) $y = (x - 3)^2$ - b) $y = x^2 - 4$ - c) $y = (x + 2)^2$ - d) $y = x^2 + 5$ - e) $y = (x + 6)^2 - 7$ - f) $y = (x - 2)^2 + 5$