Florida Algebra I EOC

1. Question 1 · Expressions, Functions & Data

   What is the value of ~(³√{64})^{2} − 81^{3/4}~?
   • A.~1~
   • B.~7~
   • C.~−1~
   • D.~−11~
   Show answer and explanation

   Answer: D. ~−11~

   The cube root of 64 is 4, and ~4^{2} = 16~. For ~81^{3/4}~, the fourth root of 81 is 3, and ~3^{3} = 27~. The value is ~16 − 27 = −11~. Taking only the roots and ignoring the powers gives ~4 − 3 = 1~, and multiplying each root by the power instead of raising it to that power gives ~8 − 9 = −1~.

2. Question 2 · Linear Relationships

   A garden arch has the shape of a parabola. On a coordinate grid measured in feet, the arch meets the ground at ~x = 0~ and ~x = 8~ and passes through the point ~(2, 6)~. Which function models the height of the arch?
   • A.~h(x) = −x^{2} + 8x~
   • B.~h(x) = [[1//2]]x^{2} − 4x~
   • C.~h(x) = −2x^{2} + 16x~
   • D.~h(x) = −[[1//2]]x^{2} + 4x~
   Show answer and explanation

   Answer: D. ~h(x) = −[[1//2]]x^{2} + 4x~

   The x-intercepts give the factored form ~h(x) = a(x − 0)(x − 8)~. Substitute ~(2, 6)~: ~6 = a(2)(−6)~, so ~a = −[[1//2]]~ and ~h(x) = −[[1//2]]x(x − 8) = −[[1//2]]x^{2} + 4x~. Inverting the division to get ~a = −2~ gives ~−2x^{2} + 16x~, which has the right intercepts but a height of 24 at ~x = 2~. Leaving ~a = −1~ without using the point gives a height of 12 there.

3. Question 3 · Linear Relationships

   A landscaper has at most $900 to spend on shrubs that cost $18 each and trees that cost $45 each. The design calls for at least 30 plants in all. Which combination is a viable solution?
   • A.20 shrubs and 12 trees
   • B.30 shrubs and 9 trees
   • C.25 shrubs and 4 trees
   • D.9 shrubs and 17 trees
   Show answer and explanation

   Answer: A. 20 shrubs and 12 trees

   The constraints are ~18s + 45t ≤ 900~ and ~s + t ≥ 30~. For 20 shrubs and 12 trees the cost is ~360 + 540 = 900~ and the total is 32 plants, so both hold. The combination of 25 shrubs and 4 trees has only 29 plants. The combination of 30 shrubs and 9 trees costs $945, which is over the budget, and 9 shrubs and 17 trees costs $927 for only 26 plants.

4. Question 4 · Linear Relationships

   Which equation represents the line that passes through the point ~(4, −3)~ and is perpendicular to the line ~2x − 5y = 10~?
   • A.~y = [[5//2]]x − 13~
   • B.~y = [[2//5]]x − [[23//5]]~
   • C.~y = −[[5//2]]x − 13~
   • D.~y = −[[5//2]]x + 7~
   Show answer and explanation

   Answer: D. ~y = −[[5//2]]x + 7~

   Solving ~2x − 5y = 10~ for ~y~ gives ~y = [[2//5]]x − 2~, so the given slope is ~[[2//5]]~ and a perpendicular slope is the opposite reciprocal, ~−[[5//2]]~. Then ~y + 3 = −[[5//2]](x − 4)~ gives ~y = −[[5//2]]x + 10 − 3 = −[[5//2]]x + 7~. Keeping the slope ~[[2//5]]~ gives a parallel line, and using ~[[5//2]]~ takes the reciprocal without changing the sign.

5. Question 5 · Expressions, Functions & Data

   The number of bacteria in a lab culture is modeled by ~B(t) = 50 · 3^{t}~, where ~t~ is the time in hours. What is the average rate of change, in bacteria per hour, from ~t = 1~ to ~t = 3~?
   • A.750
   • B.1,200
   • C.600
   • D.400
   Show answer and explanation

   Answer: C. 600

   Evaluate the endpoints: ~B(1) = 50 · 3 = 150~ and ~B(3) = 50 · 27 = 1,350~. The average rate of change is ~[[1,350 − 150//3 − 1]] = [[1,200//2]] = 600~ bacteria per hour. The value 1,200 is the change in the number of bacteria before dividing by the 2 hours, and 750 is the average of the two function values.

6. Question 6 · Expressions, Functions & Data

   What is the value of ~81^{3/4} · 8^{−2/3}~?
   • A.~[[3//4]]~
   • B.−108
   • C.~[[27//4]]~
   • D.108
   Show answer and explanation

   Answer: C. ~[[27//4]]~

   The fourth root of 81 is 3, so ~81^{3/4} = 3^{3} = 27~. The cube root of 8 is 2, so ~8^{−2/3} = [[1//2^{2}]] = [[1//4]]~, and the product is ~[[27//4]]~. Ignoring the negative sign in the exponent gives ~27 · 4 = 108~, and taking the fourth root of 81 without cubing it gives ~[[3//4]]~.

7. Question 7 · Linear Relationships

   ~x~	0	1	2	3	4
   ~f(x)~	5	−1	−3	−1	5

   The table shows some values of a quadratic function ~f~. Which equation represents ~f~?
   • A.~f(x) = (x − 2)^{2} − 3~
   • B.~f(x) = 2(x + 2)^{2} − 3~
   • C.~f(x) = 2(x − 2)^{2} + 5~
   • D.~f(x) = 2(x − 2)^{2} − 3~
   Show answer and explanation

   Answer: D. ~f(x) = 2(x − 2)^{2} − 3~

   The outputs are symmetric about ~x = 2~ (the values at 1 and 3 match, and so do the values at 0 and 4), so the vertex is ~(2, −3)~ and ~f(x) = a(x − 2)^{2} − 3~. Using ~(0, 5)~: ~5 = 4a − 3~, so ~a = 2~. Leaving ~a = 1~ gives ~f(0) = 1~, which does not match the table, and ~(x + 2)^{2}~ puts the vertex at ~x = −2~.

8. Question 8 · Linear Relationships

   A robotics club needs to buy at least 22 motors and sensors in all for a new robot. Motors cost $12 each and sensors cost $8 each, and the club can spend at most $240. Which combination is a viable solution?
   • A.9 motors and 12 sensors
   • B.16 motors and 8 sensors
   • C.13 motors and 11 sensors
   • D.10 motors and 14 sensors
   Show answer and explanation

   Answer: D. 10 motors and 14 sensors

   The constraints are ~m + s ≥ 22~ and ~12m + 8s ≤ 240~. For 10 motors and 14 sensors, ~10 + 14 = 24~ parts and ~12(10) + 8(14) = 232~ dollars, so both constraints hold. Buying 9 motors and 12 sensors costs only $204 but gives 21 parts, which is too few. The other two combinations have 24 parts but cost $256 and $244, which is over the budget.

9. Question 9 · Linear Relationships

   Which equation represents the line that passes through the point ~(4, −1)~ and is perpendicular to the line ~3x − 4y = 8~?
   • A.~y = [[4//3]]x − [[19//3]]~
   • B.~y = −[[4//3]]x + [[13//3]]~
   • C.~y = −[[3//4]]x + 2~
   • D.~y = [[3//4]]x − 4~
   Show answer and explanation

   Answer: B. ~y = −[[4//3]]x + [[13//3]]~

   Solving ~3x − 4y = 8~ for ~y~ gives ~y = [[3//4]]x − 2~, so the given slope is ~[[3//4]]~ and the perpendicular slope is the opposite reciprocal, ~−[[4//3]]~. Then ~−1 = −[[4//3]](4) + b~ gives ~b = −1 + [[16//3]] = [[13//3]]~. Keeping the slope ~[[3//4]]~ produces a parallel line, and using ~[[4//3]]~ takes the reciprocal without changing the sign.

10. Question 10 · Expressions, Functions & Data

    The graph shows the width of a beach, in feet, measured every 2 years after a first survey. What is the average rate of change in the width of the beach from year 2 to year 8?
    • A.−10 feet per year
    • B.10 feet per year
    • C.−8 feet per year
    • D.−6 feet per year
    Show answer and explanation

    Answer: A. −10 feet per year

    The graph passes through ~(2, 110)~ and ~(8, 50)~, so the average rate of change is ~[[50 − 110//8 − 2]] = [[−60//6]] = −10~ feet per year. The rate is negative because the beach is getting narrower. Dividing the change of −60 feet by 10 years instead of the 6 years in the interval gives −6, and −8 is the rate over the whole graph, from year 0 to year 10.

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