To compare the graphs that Jennifer and Matt drew, let's closely examine each equation and their transformations.
1. Rewrite Jennifer's graph for comparison:
- Jennifer's function is [tex]\( f(x) = 2x^2 + 4x \)[/tex].
- We can complete the square to rewrite this function in a form similar to Matt's function.
- First, factor out the 2 from the quadratic expression: [tex]\( f(x) = 2(x^2 + 2x) \)[/tex].
- Now complete the square inside the parenthesis:
- [tex]\( x^2 + 2x \)[/tex] can be rewritten as [tex]\( (x+1)^2 - 1 \)[/tex].
- So, [tex]\( f(x) = 2((x + 1)^2 - 1) = 2(x + 1)^2 - 2 \)[/tex].
2. Compare the rewritten forms of Jennifer's and Matt's graphs:
- Jennifer's rewritten function is [tex]\( 2(x + 1)^2 - 2 \)[/tex].
- Matt's function is [tex]\( 2(x + 1)^2 \)[/tex].
3. Identify shifts:
- Vertical shift: Jennifer's graph [tex]\( 2(x + 1)^2 - 2 \)[/tex] has a vertical shift of -2 compared to Matt's graph [tex]\( 2(x + 1)^2 \)[/tex]. This means Jennifer's graph is shifted 2 units down vertically.
- Horizontal shift: Both functions have [tex]\( (x + 1) \)[/tex] inside the square, indicating no horizontal shift between the two graphs.
4. Shape comparison:
- Both graphs have the same quadratic term [tex]\( 2(x + 1)^2 \)[/tex]. Therefore, the graphs have the same shape.
5. Vertex comparison:
- Jennifer's vertex is at [tex]\( (-1, -2) \)[/tex].
- Matt's vertex is at [tex]\( (-1, 0) \)[/tex].
6. Maximum Comparison:
- Both graphs are parabolas that open upwards, meaning each graph has a vertex that represents a minimum point, not a maximum. Therefore, there is no maximum vertex for either graph.
So, by completing the statements based on the above findings:
- Jennifer's graph is vertically shifted down by 2 units from Matt's graph.
- Jennifer's graph is horizontally shifted 0 units from Matt's graph (no horizontal shift).
- Jennifer's graph is the same shape as Matt's graph.
- Neither graph has a vertex that is a maximum because both graphs have vertices that are minimum points marked by [tex]\( (-1, -2) \)[/tex] for Jennifer's and [tex]\( (-1, 0) \)[/tex] for Matt's.
