Write the following as a function of x and y, eliminating the parameter t. Then graph the function and indicate the orientation of the curve æ = (t + 4)°, y =t – 1
Optimization
Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.
Integration
Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.
Application of Integration
In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.
Volume
In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).
Area
Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.
![**Problem Statement:**
Write the following as a function of \( x \) and \( y \), eliminating the parameter \( t \). Then graph the function and indicate the orientation of the curve.
\[
x = (t + 4)^2, \quad y = t - 1
\]
**Solution Steps:**
1. **Express \( t \) in terms of \( y \):**
From the equation \( y = t - 1 \), solve for \( t \):
\[
t = y + 1
\]
2. **Substitute \( t \) into the equation for \( x \):**
Substitute \( t = y + 1 \) into the equation \( x = (t + 4)^2 \):
\[
x = ((y + 1) + 4)^2 = (y + 5)^2
\]
3. **Final form of the function:**
The equation of the curve in terms of \( x \) and \( y \) is:
\[
x = (y + 5)^2
\]
4. **Graphing the Function:**
- The equation \( x = (y + 5)^2 \) represents a parabola that opens to the right.
- The vertex of this parabola is at \( (0, -5) \).
- The parabola will have a vertical line of symmetry along \( y = -5 \).
**Orientation of the Curve:**
- As \( t \) increases from negative infinity to positive infinity, \( y = t - 1 \) will also increase from negative infinity.
- Consequently, as \( y \) increases, \( x = (y + 5)^2 \) increases, indicating the curve moves to the right.
In summary, the solution has transformed the parametric equations into the Cartesian form, and the resulting parabola opens to the right with a vertex at \( (0, -5) \). The orientation indicates movement from left to right as \( t \) increases.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F09ba4550-40f8-48e9-8878-785c6fa72e76%2Fba71acd5-99a0-4482-9102-ebf8105ee685%2Fwanj78j_processed.png&w=3840&q=75)
Definition used -
Converting from parametric to cartesian form-
There are two equations for x and y in terms of t, we solve for t from one equation and substitute it in other equation so t will eliminate and we get one equation in x and y. That is called the cartesian equation.
Given -
Substituting the value of t in first equation, We got -
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