Polynomial Functions and Their Graphs: Algebra 2 Math Test Answer Key – PDF Download

Question 1: Degree and end behavior (2 points)

1. Degree \(5\), leading coefficient \(-3\).
2. Odd degree, negative leading coefficient: rises on the left and falls on the right.

Scoring: 1 point for each part. Part a needs both values.

Question 2: Zeros and multiplicity (2 points)

1. \(x=0\) (multiplicity \(1\)), \(x=2\) (multiplicity \(2\)), \(x=-5\) (multiplicity \(3\)).
2. Degree \(1+2+3=\) \(6\). It crosses at \(0\), touches and turns at \(2\), and flattens and crosses at \(-5\).

Scoring: 1 point for each part. In part b, the degree and at least two correct behaviors are needed.

Question 3: Factor by grouping (2 points)

1. \(x^{2}(x-4)-9(x-4)=(x-4)(x^{2}-9)=\) \((x-4)(x-3)(x+3)\).
2. \(x=4,\ 3,\ -3\). Check: \(g(3)=27-36-27+36=0\).

Scoring: 1 point for each part.

Question 4: Equation from a graph (3 points)

1. \(x=-1\) (multiplicity \(1\), crosses) and \(x=2\) (multiplicity \(2\), touches).
2. \(f(x)=a(x+1)(x-2)^{2}\) and \(f(0)=a(1)(4)=-8\), so \(a=-2\).
3. \(f(x)=-2(x+1)(x-2)^{2}\). Check: \(f(1)=-2(2)(1)=-4\).

Scoring: 1 point for each part. Allow follow-through from part a when computing the leading coefficient.

Question 5: Sketch a graph (3 points)

Zeros: \(-2\), \(0\) and \(3\), each crossing. Degree \(3\) with positive leading coefficient: falls on the left, rises on the right. Test values: \(f(-1)=4\), \(f(1)=-6\), \(f(2)=-8\).

Scoring: 1 point for the three zeros labeled; 1 point for correct end behavior; 1 point for a smooth curve with the right signs between the zeros.

Question 6: Turning points (2 points)

1. \(6-1=\) \(5\).
2. The graph has \(3\) turning points, so the degree is at least \(4\). Both ends rise, so the degree is even and the leading coefficient is positive. Least degree \(4\).

Scoring: 1 point for each part. Part b needs both the degree and the sign.

Question 7: Trail elevation (3 points)

1. \(e(x)=0\) at \(0,\ 2\) and \(6\) miles.
2. The table values are negative from \(x=3\) to \(x=5\), and \(e\) is zero at \(2\) and \(6\): between \(2\) and \(6\) miles.
3. \(-16\) feet at mile \(4\).

Scoring: 1 point for each part.

Question 8: Open box (3 points)

1. Length \(15-2x\), width \(9-2x\), height \(x\): \(V(x)=x(9-2x)(15-2x)\).
2. The width \(9-2x\) must be positive: \(0 < x < 4.5\).
3. \(V(2)=2(5)(11)=110\) and \(V(3)=3(3)(9)=81\). \(x=2\) gives the greater volume.

Scoring: 1 point for each part.

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