2. Determine the equation of the parabola after being transformed from \( f(x)=x^{2} \) by a vertical stretch of factor 4 , and a horizontal stretch of factor \( \frac{1}{3} \) reflected on the \( y \)-axis, and translations of 4 units to the right and 3 units up.

To determine the equation of the parabola after the given transformations, we'll apply each transformation step by step to the original function \( f(x) = x^2 \). 1. **Vertical Stretch by a Factor of 4:** A vertical stretch by a factor of 4 means that the \( y \)-values will be multiplied by 4. The equation becomes: \[ y = 4(x^2) \] 2. **Horizontal Stretch by a Factor of \( \frac{1}{3} \):** A horizontal stretch by a factor of \( \frac{1}{3} \) means that the \( x \)-values will be multiplied by 3. To account for this, we replace \( x \) with \( \frac{x}{3} \) in the equation: \[ y = 4\left(\left(\frac{x}{3}\right)^2\right) \] Simplifying this gives us: \[ y = 4\left(\frac{x^2}{9}\right) \] \[ y = \frac{4}{9}x^2 \] 3. **Reflection on the \( y \)-axis:** Reflecting the graph on the \( y \)-axis means that we replace \( x \) with \( -x \) in the equation: \[ y = \frac{4}{9}(-x)^2 \] Since \( (-x)^2 \) is equal to \( x^2 \), the equation remains the same: \[ y = \frac{4}{9}x^2 \] 4. **Translation 4 Units to the Right:** To translate the graph 4 units to the right, we replace \( x \) with \( x - 4 \) in the equation: \[ y = \frac{4}{9}(x - 4)^2 \] 5. **Translation 3 Units Up:** To translate the graph 3 units up, we add 3 to the \( y \)-value in the equation: \[ y = \frac{4}{9}(x - 4)^2 + 3 \] So, the final equation of the parabola after all the transformations is: \[ y = \frac{4}{9}(x - 4)^2 + 3 \]