# **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: - **Linear Graphing:** Understanding the slope-intercept form and how linear functions represent straight lines. - **Quadratic Analysis:** Identifying the direction of a parabola (upward/downward) based on the coefficient $a$. - **Vertex and Symmetry:** Calculating the turning point (vertex) and the axis of symmetry for parabolic functions. - **Intercepts:** Calculating the points where a graph crosses the $x$-axis (roots) and $y$-axis. - **Graphical Solutions:** Determining the solution set of a system of equations by finding the coordinates where their graphs intersect. - **Absolute Value Functions:** Graphing modulus functions which typically result in "V" shaped trajectories. - **Real-world Modeling:** Using functions to model economic equilibrium and physics-based trajectories. --- # **Important Formulas** Below are the key formulas used in this exercise: | Description | Formula | | :--- | :--- | | **Slope-Intercept Form** | $y = mx + c$ | | **Slope Formula** | $m = \frac{y_2 - y_1}{x_2 - x_1}$ | | **Quadratic Standard Form** | $f(x) = ax^2 + bx + c$ | | **Axis of Symmetry** | $x = -\frac{b}{2a}$ | | **Quadratic Formula** | $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ | | **Vertex $x$-coordinate** | $x = \frac{x_1 + x_2}{2}$ | --- # **Summary** This exercise focuses on the **visual representation of algebraic functions**. The primary learning objective is to transition from abstract equations to geometric shapes on a Cartesian plane. * **Key Learnings:** Mastery of plotting via tables of values and identifying critical points such as vertices and intercepts. * **Difficulties:** Students often find calculating the vertex of a quadratic and determining the precise points of intersection between two curves challenging. * **Strategies:** Always identify the type of function first (Linear, Quadratic, Cubic, etc.) to predict the shape of the graph. For quadratics, finding the vertex first provides a central point to build the rest of your table.