# **Exercise Questions** All questions in this exercise are listed below. Click on a question to view its solution. --- # **Key Concepts** This exercise focuses on the following concepts: - **Vertical Translation:** Shifting the graph up or down by adding or subtracting a constant *outside* the function. - **Horizontal Translation:** Shifting the graph left or right by adding or subtracting a constant *inside* the function argument. - **Vertical Stretching and Compression:** Changing the steepness of a graph by multiplying the function by a factor $a$. - **Horizontal Stretching and Compression:** Affecting the width of a graph by multiplying the input $x$ by a factor $c$. - **Reflection:** Flipping the graph across the x-axis (using a negative sign outside the function). - **Parent Functions:** Understanding the base shapes of $|x|$, $x^2$, $x^3$, and $\sqrt{x}$. Understanding these concepts is crucial for solving the exercise questions effectively. --- # **Important Formulas** Below are the key formulas used in this exercise: | Transformation Type | Algebraic Form | Direction/Effect | | :--- | :--- | :--- | | **Vertical Shift** | $y = f(x) \pm k$ | $+k$: Up, $-k$: Down | | **Horizontal Shift** | $y = f(x \pm h)$ | $+h$: Left, $-h$: Right | | **Vertical Scaling** | $y = a \cdot f(x)$ | $a > 1$: Stretch, $0 < a < 1$: Compress | | **Horizontal Scaling** | $y = f(c \cdot x)$ | $c > 1$: Compress, $0 < c < 1$: Stretch | | **Reflection (x-axis)** | $y = -f(x)$ | Flips graph upside down | --- # **Summary** This exercise explores **Function Transformations**, a powerful tool that allows mathematicians to sketch complex graphs by identifying a "parent" function and applying specific shifts or scales. * **Key Learnings:** Students learn the "counter-intuitive" nature of horizontal shifts (where $x+2$ moves the graph in the negative direction) compared to vertical shifts. * **Strategies:** A useful strategy is to transform the "anchor point" or vertex of the parent function first (e.g., $(0,0)$ for $x^2$), and then apply the scaling/reflection to the rest of the points. * **Verification:** Using a graphical calculator is recommended to verify how changing a single constant in the equation physically moves the curve on the Cartesian plane.