Graph The Parametric Equations $x(t) = 2t^2 - 3$ And $y(t) = T^2 + 1$.Which Of The Following Sketches Results?A. Quadratic B. Linear C. Quartic D. Exponential

Introduction

Parametric equations are a powerful tool in mathematics, allowing us to describe complex curves and surfaces using simple equations. In this article, we will explore the parametric equations $x(t) = 2t^2 - 3$ and $y(t) = t^2 + 1$, and determine which of the given sketches results from graphing these equations.

Understanding Parametric Equations

Parametric equations are equations that describe the coordinates of a point on a curve or surface in terms of a parameter, often denoted as $t$. In this case, we have two parametric equations:

$x(t) = 2t^2 - 3$ $y(t) = t^2 + 1$

These equations describe the $x$ and $y$ coordinates of a point on a curve as a function of the parameter $t$.

Graphing the Parametric Equations

To graph the parametric equations, we need to find the values of $x$ and $y$ for different values of $t$. We can do this by plugging in different values of $t$ into the equations and plotting the resulting points.

Let's start by finding the values of $x$ and $y$ for $t = 0$.

$x(0) = 2(0)^2 - 3 = -3$ $y(0) = (0)^2 + 1 = 1$

So, the point $(x, y) = (-3, 1)$ is on the curve for $t = 0$.

Next, let's find the values of $x$ and $y$ for $t = 1$.

$x(1) = 2(1)^2 - 3 = -1$ $y(1) = (1)^2 + 1 = 2$

So, the point $(x, y) = (-1, 2)$ is on the curve for $t = 1$.

We can continue this process for different values of $t$ to get a sense of the shape of the curve.

Analyzing the Graph

As we plot the points for different values of $t$, we notice that the curve is a parabola that opens upwards. The vertex of the parabola is at the point $(x, y) = (-3, 1)$, which we found earlier.

The equation of the parabola can be written in the form $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In this case, we have:

$y = t^2 + 1$ $x = 2t^2 - 3$

We can substitute the expression for $x$ into the equation for $y$ to get:

$y = (x + 3)^2$

This is the equation of a parabola that opens upwards, with its vertex at the point $(x, y) = (-3, 1)$.

Conclusion

Based on our analysis, we can conclude that the graph of the parametric equations $x(t) = 2t^2 - 3$ and $y(t) = t^2 + 1$ is a parabola that opens upwards. The vertex of the parabola is at the point $(x, y) = (-3, 1)$.

Answer

The correct answer is:

A. Quadratic

Discussion

Parametric equations are a powerful tool in mathematics, allowing us to describe complex curves and surfaces using simple equations. In this article, we explored the parametric equations $x(t) = 2t^2 - 3$ and $y(t) = t^2 + 1$, and determined which of the given sketches results from graphing these equations.

We found that the graph of the parametric equations is a parabola that opens upwards, with its vertex at the point $(x, y) = (-3, 1)$. This is a quadratic curve, which is a type of curve that can be described by a quadratic equation.

Key Takeaways

• Parametric equations are equations that describe the coordinates of a point on a curve or surface in terms of a parameter.
• The graph of a parametric equation can be a complex curve or surface.
• Quadratic curves are a type of curve that can be described by a quadratic equation.
• The vertex of a parabola is the point where the curve changes direction.

Further Reading

If you want to learn more about parametric equations and quadratic curves, here are some additional resources:

• Khan Academy: Parametric Equations
• Mathway: Quadratic Equations
• Wolfram Alpha: Parametric Equations

References

• [1] "Parametric Equations" by Khan Academy
• [2] "Quadratic Equations" by Mathway
• [3] "Parametric Equations" by Wolfram Alpha
  Q&A: Graphing Parametric Equations =====================================

Frequently Asked Questions

Q: What are parametric equations?

A: Parametric equations are equations that describe the coordinates of a point on a curve or surface in terms of a parameter, often denoted as $t$.

Q: How do I graph parametric equations?

A: To graph parametric equations, you need to find the values of $x$ and $y$ for different values of $t$. You can do this by plugging in different values of $t$ into the equations and plotting the resulting points.

Q: What is the difference between parametric equations and Cartesian equations?

A: Parametric equations describe the coordinates of a point on a curve or surface in terms of a parameter, while Cartesian equations describe the coordinates of a point on a curve or surface using the standard $x$ and $y$ coordinates.

Q: Can I use parametric equations to graph a circle?

A: Yes, you can use parametric equations to graph a circle. For example, the parametric equations $x(t) = \cos(t)$ and $y(t) = \sin(t)$ describe a circle with a radius of 1.

Q: How do I determine the type of curve that a parametric equation represents?

A: To determine the type of curve that a parametric equation represents, you need to analyze the equation and determine the type of curve that it describes. For example, if the equation is a quadratic equation, it represents a parabola.

Q: Can I use parametric equations to graph a 3D surface?

A: Yes, you can use parametric equations to graph a 3D surface. For example, the parametric equations $x(t, u) = t^2 + u^2$, $y(t, u) = t^2 - u^2$, and $z(t, u) = t + u$ describe a 3D surface.

Q: How do I find the vertex of a parabola described by parametric equations?

A: To find the vertex of a parabola described by parametric equations, you need to analyze the equation and determine the vertex of the parabola. For example, if the equation is $y = t^2 + 1$, the vertex of the parabola is at the point $(x, y) = (-3, 1)$.

Q: Can I use parametric equations to graph a function of two variables?

A: Yes, you can use parametric equations to graph a function of two variables. For example, the parametric equations $x(t, u) = t^2 + u^2$ and $y(t, u) = t^2 - u^2$ describe a function of two variables.

Q: How do I determine the domain and range of a parametric equation?

A: To determine the domain and range of a parametric equation, you need to analyze the equation and determine the values of $t$ that are valid and the corresponding values of $x$ and $y$.

Q: Can I use parametric equations to graph a curve with a hole or a gap?

A: Yes, you can use parametric equations to graph a curve with a hole or a gap. For example, the parametric equations $x(t) = \cos(t)$ and $y(t) = \sin(t)$ describe a circle with a hole at the point $(x, y) = (0, 0)$.

Conclusion

Parametric equations are a powerful tool in mathematics, allowing us to describe complex curves and surfaces using simple equations. By understanding how to graph parametric equations, we can gain a deeper understanding of the underlying mathematics and apply it to a wide range of problems.

Key Takeaways

• Parametric equations describe the coordinates of a point on a curve or surface in terms of a parameter.
• The graph of a parametric equation can be a complex curve or surface.
• Quadratic curves are a type of curve that can be described by a quadratic equation.
• The vertex of a parabola is the point where the curve changes direction.

Further Reading

If you want to learn more about parametric equations and quadratic curves, here are some additional resources:

• Khan Academy: Parametric Equations
• Mathway: Quadratic Equations
• Wolfram Alpha: Parametric Equations

References

• [1] "Parametric Equations" by Khan Academy
• [2] "Quadratic Equations" by Mathway
• [3] "Parametric Equations" by Wolfram Alpha