Sample Problems for “Problem Solving with Quadratic Equations” Lesson KEY

(1) A signal flare is fired into the air from a boat. The height of the flare in feet t seconds after it is fired is h = -16t2 + 160t.

(a) How high will the flare travel? When will it reach its maximum height? 400 feet, 5 seconds (b) When will the flare hit the water? 10 seconds (c) Explain how your answers to parts a and b could be found in a table and a graph of the equation. In a graph, the maximum point represents the maximum height of the flare, and the right hand x-intercept represents the point at which the flare hits the water. In a table, the entry when the height is its greatest represents the maximum height reached by the flare, and the entry for when the height is once again 0 represents the point at which the flare hits the water.

(2) A square has sides of length x centimeters. A new rectangle is created by increasing one dimension of the square by 3 centimeters. (a) How much greater is the area of the new rectangle than the area of the square? 3x (b) Write two equations for the area of the new rectangle. A = x(x + 3) A = x2 + 3x (c) Graph both equations on your calculator and copy the graphs onto your paper. Describe the shapes of the graphs. How do they compare?

They form the same parabola.

(3) The dot patterns below represent the first four square numbers 1, 4, 9, and 16.

1st 2nd 3rd 4th (a) What are the next two square numbers? 25, 36 (b) Write an equation for calculating the nth square number. s = n2 (c) Make a table and a graph of the first ten square numbers. Describe the pattern of change from one square number to the next. Term Number Number of Dots 1 1 2 4 3 9 4 16 5 25 6 36 7 49 8 64 9 81 10 100

The square numbers increase by consecutive odd integers beginning with 3 (3, 5, 7, …).

(4) A cube with edges of length 2 centimeters is built from centimeter cubes. If you paint the faces of this cube and then break it into centimeter cubes, how many cubes will be painted on three faces? How many will be painted on two faces? On one face? How many will be unpainted?

(a) Organize your data into a table like the one below.

Edge length Number of Number of centimeter cubes painted on of large cube cm cubes 3 faces 2 faces 1 face 0 faces 2 8 8 0 0 0 3 27 8 12 6 1 4 64 8 24 24 8 5 125 8 36 54 27 6 216 8 48 96 64

(b) Study the patterns in the table. Look for a relationship between the edge length of the large cube and the number of centimeter cubes. Tell whether the pattern change is linear, quadratic, exponential, or none of these. It would be none of these. The relationship involves cubing the length of the cube’s edge. (c) Look for relationships between the edge length of the large cube and the number of centimeter cubes painted on three faces, two faces, one face, and zero faces. Describe each relationship you find and tell whether the pattern of change is linear, quadratic, exponential, or none of these. 3 faces: Linear [y=8 regardless] 1 face: Quadratic [y = 6(x-2)2] 2 faces: Linear [y = 12(x-2)] 0 faces: None of these [ y = (x-2)3]