Final answer:
The distance d from point P to the origin on the graph y=x²−7 is given by the function d(x) = √(x² + (x²−7)²). The Euclidean norm, representing this distance, is invariant under rotations of the coordinate system. Graphing various function types and solving kinematic equations can describe different motions, including a projectile's parabolic path.
Step-by-step explanation:
To find the distance d from point P=(x,y) to the origin when it is on the graph y=x²−7, we use the distance formula d = √(x² + y²). Here, we substitute y with x²−7 to get d(x) = √(x² + (x²−7)²).
To show that the distance of point P to the origin is invariant under rotations of the coordinate system, we can demonstrate that the quantity x² + y² remains the same regardless of how the coordinates are rotated since it represents the squared radius of a circle centered at the origin. This is also known as the invariance of the Euclidean norm.
With respect to graphing and projectile motion, plotting graphs of functions such as y = x, y = e⁴x, or y = (x−2)²−7 helps us analyze different types of behavior such as linear, exponential decay, or parabolic paths. In simulating the parabolic path of a projectile, we solve kinematic equations for constant acceleration to establish a quadratic relationship between x and y.