1. The problem states that the curve crosses the x-axis at (2,0), touches it at (-4,0), and crosses the y-axis at (0,-32). 2. Since the curve touches the x-axis at x = -4, this root has multiplicity 2, so the factor is $(x + 4)^2$. 3. Since the curve crosses the x-axis at x = 2, this root has multiplicity 1, so the factor is $(x - 2)$. 4. The equation of the curve is therefore $y = k(x + 4)^2(x - 2)$ where $k$ is a constant. 5. To find $k$, use the y-intercept at (0, -32): $$-32 = k(0 + 4)^2(0 - 2) = k(16)(-2) = -32k$$ 6. Solving for $k$: $$-32 = -32k \implies k = 1$$ 7. So the equation is: $$y = (x + 4)^2(x - 2)$$ 8. For part b, the curve with equation $y = f(x - 2)$ shifts the original curve 2 units to the right. 9. The new equation is: $$y = ((x - 2) + 4)^2((x - 2) - 2) = (x + 2)^2(x - 4)$$ 10. Find the new x-intercepts by setting $y=0$: $(x + 2)^2 = 0 \implies x = -2$ (touches x-axis) $x - 4 = 0 \implies x = 4$ (crosses x-axis) 11. Find the y-intercept by setting $x=0$: $$y = (0 + 2)^2(0 - 4) = 4 \times (-4) = -16$$ 12. So the curve crosses the y-axis at (0, -16). 13. Summary: - Equation: $y = (x + 4)^2(x - 2)$ - Shifted equation: $y = (x + 2)^2(x - 4)$ - New x-intercepts: (-2, 0) (touch), (4, 0) (cross) - New y-intercept: (0, -16)