Graph The Parabola: $y = 3(x + 7)^2 + 1$.Plot Five Points On The Parabola:- The Vertex- Two Points To The Left Of The Vertex- Two Points To The Right Of The VertexThen, Click On The Graph-a-function Button.

Introduction

In mathematics, a parabola is a quadratic curve that can be represented by a quadratic equation in the form of $y = ax^2 + bx + c$. In this article, we will focus on graphing the parabola $y = 3(x + 7)^2 + 1$ and plotting five points on the parabola. We will also discuss the properties of the parabola and how to identify its key features.

Understanding the Parabola Equation

The given parabola equation is $y = 3(x + 7)^2 + 1$. To understand this equation, let's break it down into its components. The equation is in the form of $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. In this case, the vertex is $(-7, 1)$.

Graphing the Parabola

To graph the parabola, we need to plot five points on the curve. The five points are:

• The vertex: $(-7, 1)$
• Two points to the left of the vertex: $(-10, 91)$ and $(-4, 9)$
• Two points to the right of the vertex: $(-6, 10)$ and $(-2, 5)$

Here are the steps to graph the parabola:

1. Plot the vertex: The vertex is the point where the parabola changes direction. In this case, the vertex is $(-7, 1)$.
2. Plot the two points to the left of the vertex: To plot these points, we need to substitute the x-values into the equation and solve for the corresponding y-values. For example, to plot the point $(-10, 91)$, we substitute $x = -10$ into the equation and solve for $y$.
3. Plot the two points to the right of the vertex: Similarly, we need to substitute the x-values into the equation and solve for the corresponding y-values.
4. Connect the points: Once we have plotted the five points, we can connect them to form the parabola.

Plotting the Points

Here are the steps to plot the five points:

• Vertex: $(-7, 1)$

  • To plot this point, we need to substitute $x = -7$ into the equation and solve for $y$.
  • $y = 3(-7 + 7)^2 + 1 = 1$
  • Therefore, the vertex is $(-7, 1)$.

• Two points to the left of the vertex: $(-10, 91)$ and $(-4, 9)$

  • To plot these points, we need to substitute the x-values into the equation and solve for the corresponding y-values.
  • For example, to plot the point $(-10, 91)$, we substitute $x = -10$ into the equation and solve for $y$.
  • $y = 3(-10 + 7)^2 + 1 = 91$
  • Therefore, the point $(-10, 91)$ is on the parabola.
  • Similarly, we can plot the point $(-4, 9)$ by substituting $x = -4$ into the equation and solving for $y$.
  • $y = 3(-4 + 7)^2 + 1 = 9$
  • Therefore, the point $(-4, 9)$ is on the parabola.

• Two points to the right of the vertex: $(-6, 10)$ and $(-2, 5)$

  • To plot these points, we need to substitute the x-values into the equation and solve for the corresponding y-values.
  • For example, to plot the point $(-6, 10)$, we substitute $x = -6$ into the equation and solve for $y$.
  • $y = 3(-6 + 7)^2 + 1 = 10$
  • Therefore, the point $(-6, 10)$ is on the parabola.
  • Similarly, we can plot the point $(-2, 5)$ by substituting $x = -2$ into the equation and solving for $y$.
  • $y = 3(-2 + 7)^2 + 1 = 5$
  • Therefore, the point $(-2, 5)$ is on the parabola.

Graphing the Parabola

Once we have plotted the five points, we can connect them to form the parabola. The parabola is a smooth, continuous curve that passes through the five points.

Properties of the Parabola

The parabola has several key properties that can be identified from the equation. These properties include:

• Vertex: The vertex is the point where the parabola changes direction. In this case, the vertex is $(-7, 1)$.
• Axis of Symmetry: The axis of symmetry is the vertical line that passes through the vertex. In this case, the axis of symmetry is $x = -7$.
• Direction: The parabola opens upwards, indicating that it is a minimum point.

Conclusion

Introduction

In our previous article, we graphed the parabola $y = 3(x + 7)^2 + 1$ and discussed its properties. In this article, we will answer some frequently asked questions about graphing parabolas.

Q: What is a parabola?

A parabola is a quadratic curve that can be represented by a quadratic equation in the form of $y = ax^2 + bx + c$. It is a U-shaped curve that can open upwards or downwards.

Q: How do I graph a parabola?

To graph a parabola, you need to follow these steps:

1. Write the equation: Write the equation of the parabola in the form of $y = ax^2 + bx + c$.
2. Find the vertex: Find the vertex of the parabola by using the formula $x = -\frac{b}{2a}$.
3. Plot the vertex: Plot the vertex on the coordinate plane.
4. Plot two points to the left and right of the vertex: Plot two points to the left and right of the vertex by substituting the x-values into the equation and solving for the corresponding y-values.
5. Connect the points: Connect the points to form the parabola.

Q: What is the vertex of a parabola?

The vertex of a parabola is the point where the parabola changes direction. It is the minimum or maximum point of the parabola.

Q: How do I find the axis of symmetry of a parabola?

To find the axis of symmetry of a parabola, you need to use the formula $x = -\frac{b}{2a}$. This will give you the x-coordinate of the axis of symmetry.

Q: What is the direction of a parabola?

The direction of a parabola is determined by the sign of the coefficient of the $x^2$ term. If the coefficient is positive, the parabola opens upwards. If the coefficient is negative, the parabola opens downwards.

Q: How do I graph a parabola that opens downwards?

To graph a parabola that opens downwards, you need to follow the same steps as before. However, you need to make sure that the coefficient of the $x^2$ term is negative.

Q: Can a parabola have more than one vertex?

No, a parabola cannot have more than one vertex. The vertex is a unique point on the parabola.

Q: Can a parabola be a straight line?

No, a parabola cannot be a straight line. A parabola is a curved shape that is defined by a quadratic equation.

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

To graph a parabola with a horizontal axis of symmetry, you need to follow the same steps as before. However, you need to make sure that the axis of symmetry is a horizontal line.

Conclusion

In this article, we answered some frequently asked questions about graphing parabolas. We discussed the properties of parabolas, including the vertex, axis of symmetry, and direction. We also provided some tips and tricks for graphing parabolas.