Graph The Parabola: ${ Y = (x-2)^2 + 3 }$Plot Five Points On The Parabola, Including The Vertex And Two Points To The Left.

Introduction

In mathematics, a parabola is a quadratic curve that is U-shaped and has a single turning point, known as the vertex. The equation of a parabola can be written in the form $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In this article, we will focus on graphing the parabola $y = (x-2)^2 + 3$, which is a specific example of a parabola with a vertex at $(2, 3)$.

Understanding the Equation

The equation $y = (x-2)^2 + 3$ represents a parabola that opens upwards, with its vertex at $(2, 3)$. The term $(x-2)^2$ represents a quadratic expression that is squared, which means it is always non-negative. When this expression is added to $3$, the result is a parabola that has a minimum value of $3$ at the vertex.

Graphing the Parabola

To graph the parabola, we need to plot five points on the curve, including the vertex and two points to the left. We can start by finding the coordinates of the vertex, which is given by the equation $(2, 3)$. This means that the vertex is located at the point $(2, 3)$ on the coordinate plane.

Plotting the Vertex

The vertex of the parabola is located at $(2, 3)$. To plot this point, we need to locate the x-coordinate $2$ on the x-axis and the y-coordinate $3$ on the y-axis. The point of intersection between the x-axis and the y-axis is the vertex of the parabola.

Plotting Points to the Left

To plot points to the left of the vertex, we need to find the x-coordinates of these points. Since the parabola opens upwards, the x-coordinates of the points to the left of the vertex will be less than $2$. Let's choose two x-coordinates, $x = 0$ and $x = 1$, and find the corresponding y-coordinates.

Finding the Y-Coordinates

To find the y-coordinates of the points, we need to substitute the x-coordinates into the equation of the parabola. For the point with x-coordinate $0$, we have:

$$y = (0-2)^2 + 3 = 4 + 3 = 7$$

For the point with x-coordinate $1$, we have:

$$y = (1-2)^2 + 3 = 1 + 3 = 4$$

Plotting the Points

Now that we have the coordinates of the points, we can plot them on the coordinate plane. The point with x-coordinate $0$ and y-coordinate $7$ is located at $(0, 7)$. The point with x-coordinate $1$ and y-coordinate $4$ is located at $(1, 4)$.

Plotting the Parabola

To plot the parabola, we need to connect the points we have plotted. Since the parabola opens upwards, the points we have plotted will form a U-shaped curve. The vertex of the parabola is located at $(2, 3)$, and the points to the left of the vertex are located at $(0, 7)$ and $(1, 4)$.

Conclusion

In this article, we have graphed the parabola $y = (x-2)^2 + 3$, which is a specific example of a parabola with a vertex at $(2, 3)$. We have plotted five points on the curve, including the vertex and two points to the left, and connected them to form a U-shaped curve. This graph represents a parabola that opens upwards, with its vertex at $(2, 3)$.

Key Takeaways

• The equation $y = (x-2)^2 + 3$ represents a parabola that opens upwards, with its vertex at $(2, 3)$.
• The vertex of the parabola is located at $(2, 3)$.
• The points to the left of the vertex are located at $(0, 7)$ and $(1, 4)$.
• The parabola opens upwards, with its vertex at $(2, 3)$.

Further Reading

For more information on graphing parabolas, see the following resources:

• Graphing Quadratic Functions
• Parabolas
• Graphing Parabolas
  Graphing the Parabola: A Comprehensive Guide =====================================================

Q&A: Graphing the Parabola

Q: What is a parabola?

A: A parabola is a quadratic curve that is U-shaped and has a single turning point, known as the vertex. The equation of a parabola can be written in the form $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point where the parabola changes direction. It is the lowest or highest point on the parabola, depending on whether the parabola opens upwards or downwards.

Q: How do I find the vertex of a parabola?

A: To find the vertex of a parabola, you need to find the x-coordinate of the vertex, which is given by the equation $x = -\frac{b}{2a}$. Once you have the x-coordinate, you can substitute it into the equation of the parabola to find the y-coordinate.

Q: How do I graph a parabola?

A: To graph a parabola, you need to plot five points on the curve, including the vertex and two points to the left and two points to the right. You can use the equation of the parabola to find the coordinates of these points.

Q: What are the key features of a parabola?

A: The key features of a parabola include:

• The vertex: The point where the parabola changes direction.
• The axis of symmetry: The line that passes through the vertex and is perpendicular to the x-axis.
• The x-intercepts: The points where the parabola intersects the x-axis.
• The y-intercept: The point where the parabola intersects the y-axis.

Q: How do I determine the direction of a parabola?

A: To determine the direction of a parabola, you need to look at the coefficient of the $x^2$ term in the equation of the parabola. If the coefficient is positive, the parabola opens upwards. If the coefficient is negative, the parabola opens downwards.

Q: What are some common types of parabolas?

A: Some common types of parabolas include:

• Upward-opening parabolas: Parabolas that open upwards, with a vertex at the lowest point.
• Downward-opening parabolas: Parabolas that open downwards, with a vertex at the highest point.
• Horizontal parabolas: Parabolas that are horizontal, with a vertex at the highest or lowest point.

Q: How do I graph a parabola with a horizontal axis of symmetry?

A: To graph a parabola with a horizontal axis of symmetry, you need to plot the points on the curve, including the vertex and two points to the left and two points to the right. You can use the equation of the parabola to find the coordinates of these points.

Q: What are some real-world applications of parabolas?

A: Some real-world applications of parabolas include:

• Designing satellite dishes and antennas
• Creating parabolic mirrors for telescopes and solar concentrators
• Modeling the trajectory of projectiles and rockets
• Designing roller coasters and other amusement park rides

Conclusion

In this article, we have answered some common questions about graphing parabolas. We have discussed the key features of a parabola, including the vertex, axis of symmetry, x-intercepts, and y-intercept. We have also discussed how to determine the direction of a parabola and how to graph a parabola with a horizontal axis of symmetry. Finally, we have discussed some real-world applications of parabolas.