1. The problem involves understanding and comparing different functions: a linear function $y = ax + b$, a reciprocal function $y = \frac{1}{x}$, and a quadratic function described by a parabola opening upwards centered on the y-axis. 2. Given $b=0$, the linear function simplifies to $y = ax$. 3. The reciprocal function $y = \frac{1}{x}$ is undefined at $x=0$ and has vertical and horizontal asymptotes at $x=0$ and $y=0$ respectively. 4. The quadratic function, represented by the parabola, can be generally written as $y = cx^2$ where $c > 0$ for it to open upwards and be centered on the y-axis. 5. To graph these: - The linear function $y = ax$ is a straight line through the origin with slope $a$. - The reciprocal function $y = \frac{1}{x}$ has two branches, one in the first and third quadrants. - The quadratic function $y = cx^2$ is symmetric about the y-axis and opens upwards. 6. Important rules: - For $y = ax$, the slope $a$ determines the steepness. - For $y = \frac{1}{x}$, the function is undefined at $x=0$ and has asymptotes. - For $y = cx^2$, the vertex is at the origin and the parabola is symmetric. 7. Since $b=0$, the linear function passes through the origin. 8. The graph described is a parabola opening upwards centered on the y-axis, so it matches $y = cx^2$. Final answer: The functions are $y = ax$, $y = \frac{1}{x}$, and $y = cx^2$ with $b=0$ for the linear function, and the parabola corresponds to the quadratic function $y = cx^2$.