The Nature of Solutions
WARM-UP
Dolphin jump
[tex]y=-16 x^2+32 x-10[/tex]
This parabola has $\square$ x-intercepts, representing the times when the dolphin's height above water is $\square$ feet.
Answer (1)
The problem asks for the number of x-intercepts of the parabola y = − 16 x 2 + 32 x − 10 .
Calculate the discriminant: D = b 2 − 4 a c = ( 32 ) 2 − 4 ( − 16 ) ( − 10 ) = 384 .
Since 0"> D > 0 , there are two distinct real roots.
Therefore, the parabola has 2 x-intercepts.
Explanation
Understanding the Problem We are given the equation of a dolphin's jump as y = − 16 x 2 + 32 x − 10 . We need to find the number of x -intercepts of this parabola, which represent the times when the dolphin's height above water is 0 feet. The number of x -intercepts corresponds to the number of real solutions to the equation − 16 x 2 + 32 x − 10 = 0 .
Using the Discriminant To find the number of real solutions, we can use the discriminant, which is given by D = b 2 − 4 a c , where a = − 16 , b = 32 , and c = − 10 .
Calculating the Discriminant Let's calculate the discriminant: D = ( 32 ) 2 − 4 ( − 16 ) ( − 10 ) = 1024 − 640 = 384 .
Determining the Number of x-intercepts Since 0"> D = 384 > 0 , there are two distinct real roots, which means there are two x -intercepts. Therefore, the parabola has 2 x -intercepts, representing the times when the dolphin's height above water is 0 feet.
Final Answer The parabola has 2 x -intercepts, representing the times when the dolphin's height above water is 0 feet.
Examples
Understanding the trajectory of objects, like a dolphin's jump, can be modeled using quadratic equations. The x-intercepts of the parabola represent when the object is at a certain height (in this case, 0 feet above water). This concept is applicable in various fields such as sports (analyzing the trajectory of a ball), engineering (designing projectile motion), and physics (studying the motion of objects under gravity). By analyzing the discriminant, we can determine the number of times the object reaches a specific height.
