Interpreting and Comparing Functions: Algebra 2 Math Test Answer Key – PDF Download

Algebra 2 Function Test Answer Key: questions 1–5

Question 1: Key features of a cubic (3 points)

1. \(x^{3}-3x=x(x^{2}-3)=0\) gives \(x=0,\ x=\pm\sqrt{3}\).
2. The graph rises up to \((-1,2)\), falls to \((1,-2)\) and rises again: increasing for \(x < -1\) and \(x > 1\).
3. \(\frac{f(2)-f(1)}{2-1}=\frac{2-(-2)}{1}=4\).

Scoring: (a) 1 point for all three zeros; (b) 1 point for both intervals; (c) 1 point for the correct rate of 4.

Question 2: Domain and maximum in context (2 points)

1. Both \(w\) and the length \(60-2w\) must be positive, so the domain is \(0 < w < 30\).
2. The zeros are \(w=0\) and \(w=30\), so the axis is \(w=15\). Then \(A(15)=15(30)=450\). The greatest area is 450 square feet when \(w=15\).

Scoring: (a) 1 point; (b) 1 point for the area and 1 point included for the matching width (both needed).

Question 3: Average rate of change (3 points)

1. \(f(5)=5\) and \(f(1)=-3\), so \(\frac{5-(-3)}{5-1}=\frac{8}{4}=2\). The answer is 2.
2. \(\frac{1050-250}{30-10}=\frac{800}{20}=40\) dollars per poster.
3. Between 10 and 30 posters, the total cost rises by an average of $40 for each additional poster.

Scoring: 1 point for each part; part (b) requires units; part (c) must mention the interval 10 to 30 posters.

Question 4: Complete the square (3 points)

1. \(f(x)=x^{2}-8x+16-16+3\) gives \(f(x)=(x-4)^{2}-13\).
2. The vertex is \((4,-13)\), so the minimum value is \(-13\) at \(x=4\).

Scoring: (a) 1 point for adding and subtracting 16, 1 point for the correct vertex form; (b) 1 point for both the value and its location.

Question 5: Factored form of a quadratic (2 points)

1. \(g(x)=2(x^{2}-x-6)=2(x-3)(x+2)\), so the zeros are \(x=3\) and \(x=-2\).
2. The axis is halfway between the zeros: \(x=0.5\). Then \(g(0.5)=2(0.5-3)(0.5+2)=2(-2.5)(2.5)=-12.5\). The minimum value is \(-12.5\).

Scoring: (a) 1 point for the factorization and both zeros; (b) 1 point for the axis and the minimum value.

Algebra 2 Function Test Answer Key: questions 6–8

Question 6: Exponential features (2 points)

1. \(f(0)=3^{0}-9=-8\), so the y-intercept is \((0,-8)\).
2. As \(x\) decreases, \(3^{x}\) approaches 0, so \(f(x)\) approaches \(-9\). The asymptote is \(y=-9\).

Scoring: 1 point for each part. In (a), accept −8 without the coordinates.

Question 7: Compare two functions (3 points)

1. The vertex of \(f\) is \((2,9)\), so its maximum is 9. The table of \(g\) is symmetric about \(x=1\) with top value 11, so its maximum is 11. \(g\) has the greater maximum.
2. \(f(0)=-4+9=5\) and \(g(0)=9\). \(g\) has the greater y-intercept.
3. \(f:\ \frac{9-5}{2}=2\) and \(g:\ \frac{9-9}{2}=0\). \(f\) has the greater average rate of change.

Scoring: 1 point per part; each part needs the correct function and both values.

Question 8: Periodic function (2 points)

1. The graph repeats every 12 hours. The maximum depth is 10 ft and the minimum is 2 ft.
2. \(d(0)=6\) and \(d(3)=10\), so \(\frac{10-6}{3}=\frac{4}{3}\) feet per hour.

Scoring: (a) 1 point for all three values; (b) 1 point for \(\frac{4}{3}\) (about 1.33) with units.

Scoring summary

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