Graph The Equation.$y = -\frac{1}{8} X^2 + X - 4$

Introduction

Graphing equations is a fundamental concept in mathematics, and it plays a crucial role in various fields such as physics, engineering, and economics. In this article, we will focus on graphing the equation $y = -\frac{1}{8} x^2 + x - 4$. We will break down the process into manageable steps, and by the end of this article, you will be able to graph this equation with ease.

Understanding the Equation

Before we dive into graphing the equation, let's take a closer look at its components. The given equation is a quadratic equation in the form of $y = ax^2 + bx + c$, where $a = -\frac{1}{8}$, $b = 1$, and $c = -4$. The coefficient of the $x^2$ term, $a$, determines the direction and width of the parabola, while the coefficient of the $x$ term, $b$, determines the axis of symmetry.

Step 1: Determine the Vertex

The vertex of a parabola is the point where the parabola changes direction. To find the vertex, we can use the formula $x = -\frac{b}{2a}$. Plugging in the values of $a$ and $b$, we get:

$x = -\frac{1}{2(-\frac{1}{8})} = 4$

Now that we have the x-coordinate of the vertex, we can find the y-coordinate by plugging this value back into the original equation:

$y = -\frac{1}{8}(4)^2 + 4 - 4 = -\frac{1}{8}(16) = -2$

Therefore, the vertex of the parabola is at the point $(4, -2)$.

Step 2: Determine the Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola. Since the vertex is at the point $(4, -2)$, the axis of symmetry is the line $x = 4$.

Step 3: Determine the Direction and Width of the Parabola

The direction and width of the parabola are determined by the coefficient of the $x^2$ term, $a$. Since $a = -\frac{1}{8}$, the parabola opens downward and has a width of $2\sqrt{-\frac{1}{8}} = \frac{2\sqrt{2}}{4} = \frac{\sqrt{2}}{2}$.

Step 4: Plot the Parabola

Now that we have determined the vertex, axis of symmetry, direction, and width of the parabola, we can plot the parabola. We can start by plotting the vertex at the point $(4, -2)$. Then, we can plot two points on either side of the vertex, one at a distance of $\frac{\sqrt{2}}{2}$ from the vertex and the other at a distance of $-\frac{\sqrt{2}}{2}$ from the vertex.

Step 5: Connect the Points

Once we have plotted the two points, we can connect them to form the parabola. Since the parabola opens downward, the points will be connected in a downward direction.

Conclusion

Graphing the equation $y = -\frac{1}{8} x^2 + x - 4$ requires a step-by-step approach. By determining the vertex, axis of symmetry, direction, and width of the parabola, we can plot the parabola and connect the points to form the final graph. With this guide, you should be able to graph this equation with ease.

Graph of the Equation

Here is the graph of the equation $y = -\frac{1}{8} x^2 + x - 4$:

Key Takeaways

• The vertex of a parabola is the point where the parabola changes direction.
• The axis of symmetry is a vertical line that passes through the vertex of the parabola.
• The direction and width of a parabola are determined by the coefficient of the $x^2$ term.
• To graph a parabola, we need to determine the vertex, axis of symmetry, direction, and width of the parabola.

Frequently Asked Questions

• Q: What is the vertex of the parabola? A: The vertex of the parabola is at the point $(4, -2)$.
• Q: What is the axis of symmetry of the parabola? A: The axis of symmetry of the parabola is the line $x = 4$.
• Q: What is the direction of the parabola? A: The parabola opens downward.
• Q: What is the width of the parabola? A: The width of the parabola is $\frac{\sqrt{2}}{2}$.

References

• [1] "Graphing Quadratic Equations" by Math Is Fun
• [2] "Quadratic Equations" by Khan Academy
• [3] "Graphing Quadratic Functions" by Purplemath
  Graphing the Equation: A Q&A Guide =====================================

Introduction

Graphing equations is a fundamental concept in mathematics, and it plays a crucial role in various fields such as physics, engineering, and economics. In our previous article, we discussed graphing the equation $y = -\frac{1}{8} x^2 + x - 4$. In this article, we will provide a Q&A guide to help you better understand the concept of 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 $y = ax^2 + bx + c$, 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 upward or downward.

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

A: To find the vertex of a parabola, you can use the formula $x = -\frac{b}{2a}$. This will give you the x-coordinate of the vertex. Then, you can plug this value back into the original equation to find the y-coordinate.

Q: What is the axis of symmetry of a parabola?

A: The axis of symmetry of a parabola is a vertical line that passes through the vertex of the parabola. It is the line that divides the parabola into two equal parts.

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

A: To find the axis of symmetry of a parabola, you can use the formula $x = -\frac{b}{2a}$. This will give you the x-coordinate of the vertex, which is also the x-coordinate of the axis of symmetry.

Q: What is the direction of a parabola?

A: The direction of a parabola is determined by the coefficient of the $x^2$ term, $a$. If $a$ is positive, the parabola opens upward. If $a$ is negative, the parabola opens downward.

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

A: To determine the direction of a parabola, you can look at the coefficient of the $x^2$ term, $a$. If $a$ is positive, the parabola opens upward. If $a$ is negative, the parabola opens downward.

Q: What is the width of a parabola?

A: The width of a parabola is determined by the coefficient of the $x^2$ term, $a$. It is the distance between the two points on the parabola that are farthest apart.

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

A: To determine the width of a parabola, you can use the formula $width = \frac{2\sqrt{-a}}{a}$. This will give you the width of the parabola.

Q: How do I graph a parabola?

A: To graph a parabola, you need to determine the vertex, axis of symmetry, direction, and width of the parabola. Then, you can plot the parabola using these values.

Q: What are some common mistakes to avoid when graphing a parabola?

A: Some common mistakes to avoid when graphing a parabola include:

• Not determining the vertex, axis of symmetry, direction, and width of the parabola.
• Not plotting the parabola correctly.
• Not using the correct formula to determine the vertex, axis of symmetry, direction, and width of the parabola.

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. In this article, we provided a Q&A guide to help you better understand the concept of graphing quadratic equations. We hope this guide has been helpful in answering your questions and providing you with a better understanding of graphing quadratic equations.

Frequently Asked Questions

• 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.
• Q: What is the vertex of a parabola? A: The vertex of a parabola is the point where the parabola changes direction.
• Q: How do I find the vertex of a parabola? A: To find the vertex of a parabola, you can use the formula $x = -\frac{b}{2a}$.
• Q: What is the axis of symmetry of a parabola? A: The axis of symmetry of a parabola is a vertical line that passes through the vertex of the parabola.
• Q: How do I find the axis of symmetry of a parabola? A: To find the axis of symmetry of a parabola, you can use the formula $x = -\frac{b}{2a}$.

References

• [1] "Graphing Quadratic Equations" by Math Is Fun
• [2] "Quadratic Equations" by Khan Academy
• [3] "Graphing Quadratic Functions" by Purplemath