1. **Problem 1: Function and Transformations** Given the function $$f(x) = 0.2e^{x^2} - 4$$ for $$-3 \leq x \leq 2$$. (a) Sketch the graph of $$y = f(x)$$. (b) Find the coordinates of: (i) the x-intercept; (ii) the y-intercept. (c) The graph of $$f$$ is reflected in the x-axis, then translated 1 unit right and 2 units up to get $$g$$. Find $$g(x)$$. --- 2. **Step 1: Sketching the graph** - The function is $$f(x) = 0.2e^{x^2} - 4$$. - Since $$e^{x^2}$$ is always positive and symmetric about the y-axis, the graph is symmetric. - At $$x=0$$, $$f(0) = 0.2e^0 - 4 = 0.2 - 4 = -3.8$$, the minimum point. - As $$|x|$$ increases, $$e^{x^2}$$ grows rapidly, so $$f(x)$$ increases steeply. --- 3. **Step 2: Find intercepts** (i) **x-intercept:** Solve $$f(x) = 0$$ $$0.2e^{x^2} - 4 = 0$$ $$0.2e^{x^2} = 4$$ $$e^{x^2} = \frac{4}{0.2} = 20$$ Take natural log: $$x^2 = \ln 20$$ $$x = \pm \sqrt{\ln 20}$$ Numerically, $$\ln 20 \approx 2.9957$$, so $$x \approx \pm 1.73$$ (ii) **y-intercept:** Evaluate $$f(0)$$ $$f(0) = 0.2e^0 - 4 = 0.2 - 4 = -3.8$$ So y-intercept is $$(0, -3.8)$$. --- 4. **Step 3: Find $$g(x)$$** - Reflect $$f$$ in x-axis: $$y = -f(x) = -0.2e^{x^2} + 4$$ - Translate 1 unit right: replace $$x$$ by $$x-1$$ - Translate 2 units up: add 2 So, $$g(x) = -0.2e^{(x-1)^2} + 4 + 2 = -0.2e^{(x-1)^2} + 6$$ --- **Final answers:** - x-intercepts: $$\left(\pm \sqrt{\ln 20}, 0\right) \approx (\pm 1.73, 0)$$ - y-intercept: $$(0, -3.8)$$ - $$g(x) = -0.2e^{(x-1)^2} + 6$$