Math Constructed Response Test

5. Part A: What are the zeros of the function f(x) = x^2 - 6x + 8? Explain how you determined your answer on the lines provided.

Part A: The zeros are 2 and 4. To find the zeros, set the value of the function equal to 0. Then factor the quadratic expression on the right side of the equation. Next, use the Zero Product Property to set each factor equal to 0. Then, solve each of the resulting equations for x. Those values of x are the zeros of the function.

9. How many zeros does this quadratic function have? Explain how you determined your answer. Write your answer on the lines provided. f(x) = x^2 + 15x +56

There are two zeros for the function. The zeros of the function is when the function equals zero. Set the expression equal to zero and factor the expression. Then set each factor equal to zero and solve for x.

7. Part B: Arturo made an error when finding the minimum value of the function y = x^2 - 6x + 10. His work is shown below. g(x) = x^2 - 6x + 10 g(x) = (x^2 - 6x - 9) + 10 +9 g(x) = (x - 3)^2 + 19 The vertex is (3, 19), so the minimum value is 19. Describe the error that Arturo made. Then give the correct minimum value of the function. Write your answer on the lines provided.

To complete the square, Arturo should have added 9 inside the parentheses instead of subtracting 9. And to keep the equation balanced, he should have subtracted 9 outside the parentheses instead of adding 9. The correct minimum value of the function is 1.

4. The following are the steps to construct an equilateral triangle. Determine the error in the steps. Write your answer on the lines provided.

In step 2, line segment BC cannot be drawn from endpoints A and B. Point C is the intersection of the arcs drawn from endpoints A and B. The segment BC and AC are drawn after that intersection is found.

10. Part A: Explain how you could rewrite the expression 3x + 2(x^2 - 4x +1) + 5x - 2 to write it with the fewest number of terms. Write your answer on the lines provided. Part B: How many non-zero terms does the expression from Part A rewritten with the fewest number of terms contain?

Part A: First, distribute the factor 2 to each term of the expression in parentheses. Then, combine the like terms by adding or subtracting their coefficients. Part B: The simplified expression will have only one term.

2. Explain why the formula for the area of a sector is A = nr^2θ/360, where r is the radius of the circle and θ is the measure in degrees of the central angle of the sector. Use the word pi in your explanation instead of the symbol π. Write your answer on the lines provided.

The first part of the formula shows the degree measure of the central angle of the sector divided by the degree measure of the entire circle. So, it represents the fraction of the circle that consists of the sector. The second part of the formula shows pi times the radius squared, which is the area of the entire circle. So, the product of these two parts represents the fraction of the area of the circle that is included in the sector.

1. A pyramid and a rectangular prism have congruent bases and equal heights. Write a statement comparing the volume of the figure, and explain your reasoning. Write your answer on the lines provided.

The volume of a rectangular prism is equal to the area of its base times its height, and the volume of a pyramid is equal to one-third of the area of its base times its height. Since the pyramid and the rectangular prism have congruent bases, their bases have the same area. Since the heights are congruent, they also have the same height. So, the volume of the pyramid is one-third of the volume of the rectangular prism.

6. Right △ABC with altitude BD. Prove △ABC is similar to △BDC.

△ABC is a right △ --- given Altitude BD --- given ∠BDC is a right ∠ --- definition of altitude ∠ABC ≅ ∠BDC --- all right ∠'s are congruent ∠BCD ≅ ∠BCA --- reflexive property of congruence △ABC ~ △BDC --- angle-angle (AA) similarity

3. Explain how you can determine the value of sine x. Use the word theta in your explanation instead of the symbol. Write your answer on the lines provided.

Use the Pythagorean theorem to find that the length of the third side of the triangle is 4 inches. The sine of an angle is the ratio of the length of the opposite leg compared to the length of the hypotenuse. So, the sine of the theta is equal to the ratio of 4 to 5.

8. A student draws a card from a standard deck and then draws another card without replacing the first card. Explain why the probability of picking an ace on the first draw and the probability of picking a 7 on the second draw are NOT independent events. Write your answer on the lines provided.

When the first card is drawn from the deck, there are 52 cards to choose from. Because the first card is not replaced, there are only 51 cards to choose from when the second card is drawn. So, the result of picking an ace on the first draw affects the sample space and, therefore, the probability of picking a 7 on the second draw. So, the events are not independent.