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

Question 1: Degree and end behavior (2 points)

Consider \(f(x)=-3x^{5}+2x^{3}-x+4\).

1. State the degree and the leading coefficient. (1 point)
2. Describe the end behavior. (1 point)

Question 2: Zeros and multiplicity (2 points)

Let \(f(x)=x(x-2)^{2}(x+5)^{3}\).

1. List each zero with its multiplicity. (1 point)
2. State the degree and how the graph behaves at each zero. (1 point)

Question 3: Factor by grouping (2 points)

Let \(g(x)=x^{3}-4x^{2}-9x+36\).

1. Factor \(g(x)\) completely. (1 point)
2. Find all zeros. (1 point)

Question 4: Equation from a graph (3 points)

The graph of a polynomial function of degree \(3\) is shown.

1. Name each zero and its multiplicity. (1 point)
2. Find the leading coefficient. (1 point)
3. Write the function in factored form. (1 point)

Question 5: Sketch a graph (3 points)

Sketch the graph of \(f(x)=x(x-3)(x+2)\). Label the zeros and show the end behavior.

Question 6: Turning points (2 points)

1. What is the greatest number of turning points a polynomial of degree \(6\) can have? (1 point)
2. The graph below rises on both ends. State the least possible degree and whether the leading coefficient is positive or negative. (1 point)

Question 7: Trail elevation (3 points)

A trail’s elevation in feet relative to its start is \(e(x)=x(x-2)(x-6)\) after \(x\) miles. The table shows values for whole miles.

1. At which distances is the trail level with the start? (1 point)
2. Between which distances is the trail below the start? (1 point)
3. What is the lowest elevation in the table, and at which mile? (1 point)

Question 8: Open box (3 points)

Ethan makes an open box from a \(9\)-inch by \(15\)-inch sheet by cutting a square of side \(x\) inches from each corner and folding up the sides.

1. Write a function \(V(x)\) for the volume of the box. (1 point)
2. State a sensible domain. (1 point)
3. Compare \(V(2)\) and \(V(3)\). Which cut gives the greater volume? (1 point)

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