Let's address each part step by step for both functions **(a)** and **(b)**. --- ## (a) \( y(x) = \frac{\alpha x^2}{x^2 + \beta^2} \) for \( x \ge 0 \) ### **Step (i): Plot and Print Out \( y(x) \) with \( \alpha = 50, \beta = 10, \gamma = 5 \)** #### **Explanation:** - Plug in the given values: \( \alpha = 50, \beta = 10 \). (Note: \(\gamma\) is not used in this function.) - The function becomes: \( y(x) = \frac{50 x^2}{x^2 + 100} \) - You can plot this online using Desmos, GeoGebra, or other graphing tools. - The plot will start at \( y(0) = 0 \), increase, and asymptotically approach \( \alpha = 50 \) as \( x \to \infty \). #### **End of Step (i).** --- ### **Step (ii): Location and Scale Parameters** #### **Explanation:** - **Location Parameter:** Shifts the graph left/right (on \( x \)) or up/down (on \( y \)). - **Scale Parameter:** Stretches/compresses the graph horizontally or vertically. For \( y(x) = \frac{\alpha x^2}{x^2 + \beta^2} \): - **\(\alpha\):** Scales the maximum value (vertical stretch) — **scale parameter**. - **\(\beta\):** Stretches the function horizontally (it determines where the function reaches half-maximum, etc.) — **scale parameter**. - **Neither parameter is a location parameter** in this function. #### **End of Step (ii).** --- ### **Step (iii): Hand Sketch (Description)** #### **Explanation:** - Draw the \( x \)-axis and \( y \)-axis. - The function starts at \( y(0) = 0 \). - It increases rapidly for small \( x \), then more slowly, approaching \( y = \alpha \) (here, mark a horizontal line at \( y = 50 \)). - At \( x = \beta \), \( y(\beta) = \frac{\alpha \beta^2}{\beta^2 + \beta^2} = \frac{\alpha}{2} \) (here, at \( x = 10, y = 25 \)), so place a tick at \( x = 10 \), \( y = 25 \). - Label the horizontal asymptote at \( y = \alpha \) and the halfway mark at \( x = \beta \). #### **End of Step (iii).** --- ### **Final Answer for (a):** - **(i)** Plug in the values and plot \( y(x) = \frac{50x^2}{x^2 + 100} \) for \( x \geq 0 \). - **(ii)** Both \(\alpha\) and \(\beta\) are **scale parameters**. - **(iii)** Hand sketch should show a curve from (0,0) up to \( y = 50 \) asymptote, with a tick at \( x = 10 \) (where \( y = 25 \)), and horizontal line at \( y = 50 \) (asymptote). --- ## (b) \( y(x) = \frac{1}{1 + e^{-(x - \alpha)/\beta}} \), \( x \in (-\infty, \infty) \) ### **Step (i): Plot and Print Out \( y(x) \) with \( \alpha = 50, \beta = 10 \)** #### **Explanation:** - Plug in \( \alpha = 50, \beta = 10 \): \( y(x) = \frac{1}{1 + e^{-(x - 50)/10}} \) - This is the standard sigmoid (logistic) function, centered at \( x = 50 \), with "width" determined by \( \beta = 10 \). - Plot this in Desmos, GeoGebra, etc. - The function rises from 0 (as \( x \to -\infty \)) to 1 (as \( x \to +\infty \)), with \( y(50) = 0.5 \). #### **End of Step (i).** --- ### **Step (ii): Location and Scale Parameters** #### **Explanation:** - \(\alpha\) shifts the curve left or right (center of transition), so **location parameter**. - \(\beta\) controls the steepness (horizontal stretching), so **scale parameter**. #### **End of Step (ii).** --- ### **Step (iii): Hand Sketch (Description)** #### **Explanation:** - Draw an S-shaped curve from \( y = 0 \) to \( y = 1 \). - Mark \( x = \alpha = 50 \) on the \( x \)-axis; at this point, \( y = 0.5 \). - The width of the transition is controlled by \( \beta \) (e.g., from \( x = 40 \) to \( x = 60 \), the function goes from about 0.12 to 0.88). - Indicate \( x = 50 \) (center) and show the "steepness" is related to \( \beta \). #### **End of Step (iii).** --- ### **Final Answer for (b):** - **(i)** Plug in the values and plot \( y(x) = \frac{1}{1 + e^{-(x - 50)/10}} \). - **(ii)** \(\alpha\) is a **location parameter**; \(\beta\) is a **scale parameter**. - **(iii)** Hand sketch should show an S-curve from \( y = 0 \) to \( y = 1 \), with the midpoint at \( x = 50 \) (tick mark). --- **Summary Table** | Part | Function | Plot (i) | Parameters (ii) | Sketch (iii) | |---|---|---|---|---| | (a) | \( \frac{\alpha x^2}{x^2 + \beta^2} \) | \( \alpha = 50, \beta = 10 \) | Both are **scale parameters** | Curve from (0,0) to \( y = \alpha \), mark \( x = \beta \) and \( y = \alpha \) | | (b) | \( \frac{1}{1 + e^{-(x - \alpha)/\beta}} \) | \( \alpha = 50, \beta = 10 \) | \( \alpha \): **location**, \( \beta \): **scale** | S-curve, mark \( x = \alpha \) at \( y = 0.5 \) | --- **If you need the actual graphs, use Desmos or GeoGebra to plot with the given parameter values!**