Question

The parametric equations x = square root t, y= 3t - 4 for t in [ 0,4] define a plane curve

Answer 1

The student's question involves parametric equations defining a plane curve, and not a quadratic equation. However, if presented with a quadratic equation, the solution can be found using the quadratic formula.

Step-by-step explanation:

The parametric equations x = √(t), y = 3t - 4 define a plane curve for t in the interval [0,4]. This is a set of mathematical expressions used to describe a curve in a two-dimensional plane, where t acts as the parameter. It is not a quadratic equation, which typically has the form at² + bt + c = 0. However, should you encounter a quadratic equation, the solutions can be found using the quadratic formula which is given by -b ± √(b² - 4ac) / 2a. The quadratic formula helps to solve for the values of t when the quadratic equation is set to zero.

Answer 2

a) We kindly invite you to read the explanation and check the image attached below for further details.

b) The rectangular equation is
`y = 3\cdot x^(2)-4`.

Step-by-step explanation:

a) Let
`x = √(t)` and
`y = 3\cdot t -4` for
`t\in[0,4]`. We proceed to graph the expression with the help of a graphing tool. In this case, the tool Desmos is used and the result is presented below.

b) The rectangular equation for the curve is:

`t = x^(2)` (1)

`y = 3\cdot t - 4` (2)

By applying (1) in (2), we get the following equation in rectangular form:

`y = 3\cdot x^(2)-4`