1. **Problem Statement:** Graph and compare the functions $y = x^2 + 3$, $y = (x + 3)^2$, and $y = x^2$. Identify similarities and differences. 2. **Recall the base function:** The base function is $y = x^2$, a parabola with vertex at the origin $(0,0)$, opening upwards. 3. **Analyze $y = x^2 + 3$:** This is a vertical shift of $y = x^2$ upward by 3 units. 4. **Analyze $y = (x + 3)^2$:** This is a horizontal shift of $y = x^2$ to the left by 3 units. 5. **Summary of transformations:** - $y = x^2 + 3$: vertex moves from $(0,0)$ to $(0,3)$. - $y = (x + 3)^2$: vertex moves from $(0,0)$ to $(-3,0)$. 6. **Similarities:** - All are parabolas opening upwards. - All have the same shape (same width and orientation). 7. **Differences:** - $y = x^2 + 3$ is shifted vertically. - $y = (x + 3)^2$ is shifted horizontally. 8. **Desmos function for visualization:** $$y = x^2, \quad y = x^2 + 3, \quad y = (x + 3)^2$$ Final answer: The graphs of $y = x^2 + 3$ and $y = (x + 3)^2$ are both parabolas like $y = x^2$, but the first is shifted up by 3 units, and the second is shifted left by 3 units.