Graph each function. Then identify the domain and range of the function. \( \begin{array}{lll}\text { 10. } y=-4 x^{2} & \text { 12. } f(x)=3 x^{2} \\ \text { 13. } f(x)=\frac{2}{3} x^{2} & \text { 14. } y(x)=1.5 x^{2} & \text { 15. } y=-\frac{1}{3} x^{2}\end{array} \)

Let's analyze each function one by one, graph them, and identify their domains and ranges. ### 10. \( y = -4x^2 \) **Graph:** This is a downward-opening parabola with its vertex at the origin (0, 0). **Domain:** The domain of any quadratic function is all real numbers, so: \[ \text{Domain: } (-\infty, \infty) \] **Range:** Since the parabola opens downwards, the maximum value is at the vertex (0, 0), and it goes downwards indefinitely: \[ \text{Range: } (-\infty, 0] \] --- ### 12. \( f(x) = 3x^2 \) **Graph:** This is an upward-opening parabola with its vertex at the origin (0, 0). **Domain:** The domain is all real numbers: \[ \text{Domain: } (-\infty, \infty) \] **Range:** Since the parabola opens upwards, the minimum value is at the vertex (0, 0), and it goes upwards indefinitely: \[ \text{Range: } [0, \infty) \] --- ### 13. \( f(x) = \frac{2}{3}x^2 \) **Graph:** This is also an upward-opening parabola with its vertex at the origin (0, 0), but it is wider than the previous one. **Domain:** The domain is all real numbers: \[ \text{Domain: } (-\infty, \infty) \] **Range:** The minimum value is at the vertex (0, 0), and it goes upwards indefinitely: \[ \text{Range: } [0, \infty) \] --- ### 14. \( y(x) = 1.5x^2 \) **Graph:** This is another upward-opening parabola with its vertex at the origin (0, 0). **Domain:** The domain is all real numbers: \[ \text{Domain: } (-\infty, \infty) \] **Range:** The minimum value is at the vertex (0, 0), and it goes upwards indefinitely: \[ \text{Range: } [0, \infty) \] --- ### 15. \( y = -\frac{1}{3}x^2 \) **Graph:** This is a downward-opening parabola with its vertex at the origin (0, 0), but it is wider than the first function. **Domain:** The domain is all real numbers: \[ \text{Domain: } (-\infty, \infty) \] **Range:** Since the parabola opens downwards, the maximum value is at the vertex (0, 0), and it goes downwards indefinitely: \[ \text{Range: } (-\infty, 0] \] --- ### Summary of Domains and Ranges | Function | Domain | Range | |-------------------------|-------------------|-----------------| | \( y = -4x^2 \) | \( (-\infty, \infty) \) | \( (-\infty, 0] \) | | \( f(x) = 3x^2 \) | \( (-\infty, \infty) \) | \( [0, \infty) \) | | \( f(x) = \frac{2}{3}x^2 \) | \( (-\infty, \infty) \) | \( [0, \infty) \) | | \( y(x) = 1.5x^2 \) | \( (-\infty, \infty) \) | \( [0, \infty) \) | | \( y = -\frac{1}{3}x^2 \) | \( (-\infty, \infty) \) | \( (-\infty, 0] \) | These functions are all parabolas, and their domains are all real numbers, while their ranges depend on whether they open upwards or downwards.