Given the function \( h(x)=\frac{2}{x-2}+1 \). 4.2.1 Write down the equations of the asymptotes of \( h \). 4.2.2 Determine the equation of the decreasing axis of symmetry of \( h(x-1) \) (3)

Let's analyze the given function and answer both parts of the question step by step. ### Given Function: \[ h(x) = \frac{2}{x - 2} + 1 \] ### 4.2.1 Asymptotes of \( h(x) \): **Vertical Asymptote:** The vertical asymptote occurs where the denominator is zero. \[ x - 2 = 0 \implies x = 2 \] **Horizontal Asymptote:** As \( x \) approaches infinity, the term \( \frac{2}{x - 2} \) approaches 0. \[ \lim_{x \to \pm\infty} h(x) = 1 \] **Equations of the Asymptotes:** - **Vertical Asymptote:** \( x = 2 \) - **Horizontal Asymptote:** \( y = 1 \) ### 4.2.2 Decreasing Axis of Symmetry of \( h(x - 1) \): First, let's find \( h(x - 1) \): \[ h(x - 1) = \frac{2}{(x - 1) - 2} + 1 = \frac{2}{x - 3} + 1 \] **Center of the Hyperbola:** The center is at \( (3, 1) \). **Symmetry Axes:** For the function \( h(x) = \frac{a}{x - h} + k \), the hyperbola is symmetric about two lines: 1. \( y = x - (h - k) \) (increasing symmetry axis) 2. \( y = -x + (h + k) \) (decreasing symmetry axis) Applying this to \( h(x - 1) \): \[ h = 3, \quad k = 1 \] - **Increasing Symmetry Axis:** \( y = x - 2 \) - **Decreasing Symmetry Axis:** \( y = -x + 4 \) Since the question asks for the **decreasing** axis of symmetry: **Equation of the Decreasing Axis of Symmetry:** \[ y = -x + 4 \] --- **Final Answers:** **4.2.1** - **Vertical Asymptote:** \( x = 2 \) - **Horizontal Asymptote:** \( y = 1 \) **4.2.2** - **Decreasing Axis of Symmetry of \( h(x - 1) \):** \( y = -x + 4 \)