Graph The Equation Y = − 5 ( X ) 2 + 1 Y = -5(x)^2 + 1 Y = − 5 ( X ) 2 + 1 . Which Of The Following Statements About The Graph Is Correct?A. The Parabola Will Have A Vertex At ( 1 , 0 (1,0 ( 1 , 0 ].B. The Parabola Will Open Downwards.C. The Parabola Will Have A Vertex At

Introduction

Graphing quadratic equations is a fundamental concept in mathematics, and it plays a crucial role in various fields such as physics, engineering, and economics. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. In this article, we will focus on graphing the equation $y = -5(x)^2 + 1$ and analyzing its characteristics.

Understanding the Equation

The given equation is $y = -5(x)^2 + 1$. This is a quadratic equation in the form $y = ax^2 + bx + c$, where $a = -5$, $b = 0$, and $c = 1$. The coefficient of the squared term, $a$, determines the direction of the parabola's opening. If $a$ is positive, the parabola opens upwards, and if $a$ is negative, the parabola opens downwards.

Graphing the Equation

To graph the equation $y = -5(x)^2 + 1$, we can start by identifying the vertex of the parabola. The vertex is the point on the parabola where it changes direction. Since the equation is in the form $y = ax^2 + bx + c$, we can use the formula $x = -\frac{b}{2a}$ to find the x-coordinate of the vertex.

In this case, $a = -5$ and $b = 0$, so the x-coordinate of the vertex is $x = -\frac{0}{2(-5)} = 0$. To find the y-coordinate of the vertex, we can substitute $x = 0$ into the equation: $y = -5(0)^2 + 1 = 1$. Therefore, the vertex of the parabola is at the point $(0, 1)$.

Analyzing the Graph

Now that we have identified the vertex of the parabola, we can analyze its characteristics. Since the coefficient of the squared term, $a$, is negative, the parabola opens downwards. This means that the parabola will have a minimum point at the vertex, and it will decrease as we move away from the vertex in both directions.

Conclusion

In conclusion, the graph of the equation $y = -5(x)^2 + 1$ is a parabola that opens downwards. The vertex of the parabola is at the point $(0, 1)$, and it has a minimum point at this vertex. The parabola decreases as we move away from the vertex in both directions.

Answer to the Question

Based on our analysis, we can conclude that the correct statement about the graph is:

• B. The parabola will open downwards.

Q&A: Graphing Quadratic Equations

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: What is the significance of the coefficient of the squared term, $a$, in a quadratic equation?

A: The coefficient of the squared term, $a$, determines the direction of the parabola's opening. If $a$ is positive, the parabola opens upwards, and if $a$ is negative, the parabola opens downwards.

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

A: To find the vertex of a parabola, you can use the formula $x = -\frac{b}{2a}$ to find the x-coordinate of the vertex. Then, substitute $x$ into the equation to find the y-coordinate of the vertex.

Q: What is the vertex of a parabola?

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

Q: How do I determine whether a parabola opens upwards or downwards?

A: To determine whether a parabola opens upwards or downwards, look at the coefficient of the squared term, $a$. If $a$ is positive, the parabola opens upwards, and if $a$ is negative, the parabola opens downwards.

Q: What is the significance of the y-intercept in a quadratic equation?

A: The y-intercept is the point on the parabola where it intersects the y-axis. It is the value of $y$ when $x = 0$. The y-intercept can be found by substituting $x = 0$ into the equation.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, start by identifying the vertex of the parabola. Then, use the vertex to draw the parabola. You can also use a graphing calculator or software to graph the equation.

Q: What are some common mistakes to avoid when graphing quadratic equations?

A: Some common mistakes to avoid when graphing quadratic equations include:

• Not identifying the vertex of the parabola correctly
• Not using the correct formula to find the x-coordinate of the vertex
• Not substituting $x$ into the equation to find the y-coordinate of the vertex
• Not using a graphing calculator or software to check the graph

Conclusion

In conclusion, graphing quadratic equations is a fundamental concept in mathematics, and it plays a crucial role in various fields such as physics, engineering, and economics. By understanding the characteristics of parabolas and how to graph them, you can solve a wide range of problems and make informed decisions.