Understanding Quadratic Equations Through Real World Problems

Building Understanding for a Quadratic Equation through Asking Questions about the Graph of the Height of a Ball Hit by a Baseball Player

A baseball batter pops a ball straight up A baseball batter pops a ball straight up. The height of the ball is shown as a function of time in a graph. What types of information can be learn about the height of the ball?

From the graph describe the following: The ball's initial height when it is hit by the bat The ball's maximum height The time when the ball reached it's maximum height How many times is the ball 20 feet above the ground? When?

Building Understanding for the Constants Used in the common Quadratic Equation That Represents the Particle Motion of a Rocket

A model rocket blasts off and its engine shuts down when it is 25 m above the ground. Its velocity at that time is 50 m/s. Assume that it travels straight up and that the only force acting on it is the downward pull of gravity. In the metric system, the acceleration due to gravity is 9.8 m/s2. The quadratic function h(t)=(1/2)(-9.8)t2+50t+25 describes the rocket’s projectile motion as a function of time, t. What variables are used in the equation? What do they represent? What are their units?

A model rocket blasts off and its engine shuts down when it is 25 m above the ground. Its velocity at that time is 50 m/s. Assume that it travels straight up and that the only force acting on it is the downward pull of gravity. In the metric system, the acceleration due to gravity is 9.8 m/s2. The quadratic function h(t)=(1/2)(-9.8)t2+50t+25 describes the rocket’s projectile motion as a function of time, t. What is the real world meaning for h(0)=25?

A model rocket blasts off and its engine shuts down when it is 25 m above the ground. Its velocity at that time is 50 m/s. Assume that it travels straight up and that the only force acting on it is the downward pull of gravity. In the metric system, the acceleration due to gravity is 9.8 m/s2. The quadratic function h(t)=(1/2)(-9.8)t2+50t+25 describes the rocket’s projectile motion as a function of time, t. How is the acceleration due to gravity represented in the equation? How does the equation show the force is downward?

Building Further Understanding the Analytic, Graphical and Numerical Information about a Quadratic Equation in General Form

Taylor hits a baseball and its height in the air at time x is given by the equation below, where x is measured in seconds and y is measured in feet. What do the constants represent in this equation? What additional information would a student learn from a table and a graph of this equation?

h(t)=-16t2+58t+3 x y 0.00000 3.00000 0.20000 13.96000 0.40000 23.64000 0.60000 32.04000 0.80000 39.16000 1.00000 45.00000 1.20000 49.56000 1.40000 52.84000 1.60000 54.84000 1.80000 55.56000 2.00000 55.00000 2.20000 53.16000 2.40000 50.04000 2.60000 45.64000 2.80000 39.96000 3.00000 33.00000 3.20000 24.76000 3.40000 15.24000 3.60000 4.44000 3.80000 -7.64000 How high does the baseball fly before falling back to earth? How long does it take the baseball to reach its maximum height?

h(t)=-16t2+58t+3 x y 0.00000 3.00000 0.20000 13.96000 0.40000 23.64000 0.60000 32.04000 0.80000 39.16000 1.00000 45.00000 1.20000 49.56000 1.40000 52.84000 1.60000 54.84000 1.80000 55.56000 2.00000 55.00000 2.20000 53.16000 2.40000 50.04000 2.60000 45.64000 2.80000 39.96000 3.00000 33.00000 3.20000 24.76000 3.40000 15.24000 3.60000 4.44000 3.80000 -7.64000 After the ball is hit, how long does the ball remain in flight? What domain and range make sense in this situation?

h(t)=-16t2+58t+3 x y 0.00000 3.00000 0.20000 13.96000 0.40000 23.64000 0.60000 32.04000 0.80000 39.16000 1.00000 45.00000 1.20000 49.56000 1.40000 52.84000 1.60000 54.84000 1.80000 55.56000 2.00000 55.00000 2.20000 53.16000 2.40000 50.04000 2.60000 45.64000 2.80000 39.96000 3.00000 33.00000 3.20000 24.76000 3.40000 15.24000 3.60000 4.44000 3.80000 -7.64000 When and how often is the ball 40 feet above the ground? What equation would you have to solve to find the two times?

What effect will changing the initial velocity have on the flight of the baseball? What effect will changing the initial height have on the flight of the baseball?

Building Understanding for Roots and the Vertex of a Quadratic Equation

Length Width Area 1 11 2 10 20 3 9 27 4 8 32 5 7 35 6 36 Students can record the dimensions of at least eight different rectangular regions, each with perimeter 24 meters and then find the area of the possible gardens.

What garden widths would have no area? Length Width Area 12 1 11 2 10 20 3 9 27 4 8 32 5 7 35 6 36 What garden widths would have no area? If x is a length of the garden, what expression represents the width of the garden? What expression would represent the area of the garden? x 12-x x(12-x)

Students could create a scatter plot for the width in meters vs Students could create a scatter plot for the width in meters vs. the area in square meters. Then they could be asked to describe as completely as possible some of the characteristics of the graph. Does it make sense to connect the points with a smooth curve? Compare to a loop of 24 inches of string the model the constant perimeter.

At what x value does the graph reach its largest area? What is special about this shape? Compare to the changing string.

Students could enter the area equation in their graphing calculator Students could enter the area equation in their graphing calculator. Does their equation match the data graphed in the previous step? Notice where the graph crosses the x-axis. What is the real-world meaning for these points? Suppose we changed the perimeter of the garden, how would this affect the shape of the garden when it is largest?

Three points were highlighted in this activity. The two values on the x-axis where the quadratic equation crosses the x-axis. This is where the area will be zero. The vertex is the location of the largest area. What is the relationship between the three x-values?

Recall your equation for the area was A=(x)(12-x) which is called the factored form. Expanding this form yields A = -x2+12x. Another common equation for a quadratic is the vertex form: y=a(x-h)2+k, where the vertex is (h,k). So our area would be A=-1(x-6)2 +36.

The graphs at the right show how the length, width, and area are changing. Why does the area curve have zeros at 0 and 12? Why does the area curve have a maximum at 6? Why does the area curve have symmetry about x = 6?

The Yo-Yo Warehouse uses the equation to model the relationship between income and the price for one of its top-selling yo-yos. In this model the y represents the income in dollars and x represents the selling price in dollars of one yo-yo. Create a graph of this relationship.

The Yo-Yo Warehouse uses the equation to model the relationship between income and the price for one of its top-selling yo-yos. In this model the y represents the income in dollars and x represents the selling price in dollars of one yo-yo. What would be a reasonable way to find the zeros? What do they mean?

The Yo-Yo Warehouse uses the equation to model the relationship between income and the price for one of its top-selling yo-yos. In this model the y represents the income in dollars and x represents the selling price in dollars of one yo-yo. What are the meanings of the coordinates of the vertex? How can you find these coordinates?

The Yo-Yo Warehouse uses the equation to model the relationship between income and the price for one of its top-selling yo-yos. In this model the y represents the income in dollars and x represents the selling price in dollars of one yo-yo. What is a reasonable domain and range for this relationship?

The Yo-Yo Warehouse uses the equation to model the relationship between income and the price for one of its top-selling yo-yos. In this model the y represents the income in dollars and x represents the selling price in dollars of one yo-yo. Explain the meaning of the point (5, $637.50)

Three Forms for a Quadratic Equation General: Factored: Vertex:

Suppose a parabola passes through (2,0) and (6,0) Suppose a parabola passes through (2,0) and (6,0). Suppose also that its vertex is located at (4,8). Is it possible to represent the equation in the vertex form? Is it possible to represent the equation in factored form? How are the equations related? How do the values of h and k affect the graph? How do the values or r1 and r2 affect the graph?

Bob and Scott are playing golf Bob and Scott are playing golf. Bob hits the ball and it is in flight for 3.4 seconds. Scott’s ball is in flight for 4.7 seconds. At what time does each ball reach its highest height? Can you tell whose ball goes farther or higher?