of Readiness for College and Careers
Algebra I/Math II sample question
Michelle wanted to investigate the effect on the vertex of the graph of f(x)=xB+6x when f(x) is replaced by f(x+k). Michelle graphed functions of the form f(x+k) for k=1,2,3 and 4. For each of the functions she graphed, the x-coordinate of the vertex was negative and different for each value of k, but the y-coordinate of the vertex was the same for each value of k. Michelle made three conjectures based on her results.
1. The x-coordinate of the vertex depends on the value of k.
2. The x-coordinate of the vertex is negative for all values of k.
3. The y-coordinate of the vertex is independent of the value of k.
Determine if each of Michelle's three conjectures are true. Justify each answer.
Elena had 5 beads so Damian has 5+8=13 beads. Trish has 4x13=52. Altogether, there are 5+13+52=70 beads. I know that 80 is 10 more than 70, so there were 80 total beads.
Part A: 3
Part B: Explains or shows how to use the 80 percent. Explains or shows how to use the 3:1 ratio. Provides answer of 126.
Part A: Nancy (160/5= 32 pages per day); Barbara (100/4= 25 pages per day); Colleen (54/3= 18 pages per day)
Part B: Barbara (320−100/25= 8.8 days); Nancy (480−160/32= 10.0 days); Colleen (260−54/18= 11.4 days)
Algebra I/Math II
Students will earn one reasoning point for an adequate proof that conjecture 1 is correct. For example: As k increases, the vertex moves so I know that the x coordinate of the vertex will change as I change k.
Students will earn one reasoning point for an adequate proof that conjecture 2 is incorrect. For example: I figured that when k is much less than 1, then the vertex would be positive. I solved used k= -5 and found that f(x-5)=xB-4x-5=(x-2)B-9, and the vertex is at (2,-9).
For example: f(x+k)=(x-(-k-3))B-9. Therefore, when -k-3>0, i.e., when k<-3, the x-coordinate of the vertex is -k-3, which is positive. Students will earn one reasoning point for an adequate algebraic proof based on completing the square that conjecture 3 is correct.
For example: Because the completed equation is f(x+k)= (x+ (k+3))B-9. The vertex of the graph is (-k-3,-9) and the y-coordinate of the vertex will always be -9. It is independent of k.Source: Partnership for Assessment of Readiness for College and Careers