Write An Equation In Standard Form Using The Table Below.${ \begin{array}{|c|c|c|c|c|c|} \hline x & 5 & 6 & 7 & 8 & 9 \ \hline y & 118 & 125 & 118 & 97 & 62 \ \hline \end{array} }$A. $x^2 - 12x + 23$B. $-7x^2 + 42x -

Introduction

In mathematics, a parabola is a quadratic equation that can be represented in various forms, including standard form. The standard form of a parabola is given by the equation $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In this article, we will learn how to write an equation in standard form using a given table of values.

Understanding the Table of Values

The table below represents a set of values for the variables $x$ and $y$.

x	y
5	118
6	125
7	118
8	97
9	62

Step 1: Find the Value of $a$

To find the value of $a$, we need to examine the table and look for a pattern in the values of $y$. We can see that the values of $y$ are increasing as the values of $x$ increase. This suggests that the parabola is opening upwards, and the value of $a$ is positive.

Let's calculate the value of $a$ using the first two values in the table.

$$y_1 = 118 = a(x_1)^2 + b(x_1) + c$$

$$y_2 = 125 = a(x_2)^2 + b(x_2) + c$$

Subtracting the two equations, we get:

$$7 = a(x_2^2 - x_1^2) + b(x_2 - x_1)$$

Simplifying the equation, we get:

$$7 = a(36 - 25) + b(6 - 5)$$

$$7 = 11a + b$$

Now, let's use the third value in the table to find the value of $a$.

$$y_3 = 118 = a(x_3)^2 + b(x_3) + c$$

Subtracting the first equation from the third equation, we get:

$$0 = a(x_3^2 - x_1^2) + b(x_3 - x_1)$$

Simplifying the equation, we get:

$$0 = a(49 - 25) + b(7 - 5)$$

$$0 = 24a + 2b$$

Now we have two equations with two variables:

$$7 = 11a + b$$

$$0 = 24a + 2b$$

Solving the system of equations, we get:

$$a = 1$$

Step 2: Find the Value of $b$

Now that we have the value of $a$, we can find the value of $b$ using the first two values in the table.

$$y_1 = 118 = a(x_1)^2 + b(x_1) + c$$

$$y_2 = 125 = a(x_2)^2 + b(x_2) + c$$

Substituting the value of $a$, we get:

$$118 = 1(5)^2 + b(5) + c$$

$$125 = 1(6)^2 + b(6) + c$$

Simplifying the equations, we get:

$$118 = 25 + 5b + c$$

$$125 = 36 + 6b + c$$

Subtracting the two equations, we get:

$$7 = 11 + b$$

Solving for $b$, we get:

$$b = -4$$

Step 3: Find the Value of $c$

Now that we have the values of $a$ and $b$, we can find the value of $c$ using the first value in the table.

$$y_1 = 118 = a(x_1)^2 + b(x_1) + c$$

Substituting the values of $a$ and $b$, we get:

$$118 = 1(5)^2 + (-4)(5) + c$$

Simplifying the equation, we get:

$$118 = 25 - 20 + c$$

$$118 = 5 + c$$

Solving for $c$, we get:

$$c = 113$$

Writing the Equation in Standard Form

Now that we have the values of $a$, $b$, and $c$, we can write the equation in standard form.

$$y = ax^2 + bx + c$$

Substituting the values, we get:

$$y = 1x^2 - 4x + 113$$

Simplifying the equation, we get:

$$y = x^2 - 4x + 113$$

Conclusion

In this article, we learned how to write an equation in standard form using a given table of values. We found the values of $a$, $b$, and $c$ using the table and then wrote the equation in standard form. The equation is:

$$y = x^2 - 4x + 113$$

This equation represents a parabola that opens upwards, and the value of $a$ is positive. The value of $b$ is negative, which means that the parabola is shifted to the left. The value of $c$ is positive, which means that the parabola is shifted upwards.

Discussion

The equation we found is a quadratic equation, which means that it can be represented in various forms, including standard form. The standard form of a quadratic equation is given by the equation $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

In this case, the value of $a$ is positive, which means that the parabola is opening upwards. The value of $b$ is negative, which means that the parabola is shifted to the left. The value of $c$ is positive, which means that the parabola is shifted upwards.

The equation we found can be used to model various real-world situations, such as the trajectory of a projectile or the growth of a population. It can also be used to solve problems involving quadratic equations.

References

• [1] "Quadratic Equations" by Math Open Reference
• [2] "Parabolas" by Math Is Fun
• [3] "Quadratic Formula" by Wolfram MathWorld

Additional Resources

• [1] "Quadratic Equations" by Khan Academy
• [2] "Parabolas" by Brilliant
• [3] "Quadratic Formula" by MIT OpenCourseWare
  Frequently Asked Questions (FAQs) =====================================

Q: What is a parabola?

A: A parabola is a quadratic equation that can be represented in various forms, including standard form. It is a U-shaped curve that opens upwards or downwards.

Q: What is the standard form of a parabola?

A: The standard form of a parabola is given by the equation $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: How do I find the value of $a$ in a parabola?

A: To find the value of $a$, you need to examine the table of values and look for a pattern in the values of $y$. You can use the first two values in the table to find the value of $a$.

Q: How do I find the value of $b$ in a parabola?

A: To find the value of $b$, you need to use the values of $a$ and the first two values in the table. You can subtract the two equations to find the value of $b$.

Q: How do I find the value of $c$ in a parabola?

A: To find the value of $c$, you need to use the values of $a$ and $b$, and the first value in the table. You can substitute the values of $a$ and $b$ into the equation and solve for $c$.

Q: What is the significance of the value of $a$ in a parabola?

A: The value of $a$ determines the direction and width of the parabola. If $a$ is positive, the parabola opens upwards. If $a$ is negative, the parabola opens downwards.

Q: What is the significance of the value of $b$ in a parabola?

A: The value of $b$ determines the horizontal shift of the parabola. If $b$ is positive, the parabola is shifted to the right. If $b$ is negative, the parabola is shifted to the left.

Q: What is the significance of the value of $c$ in a parabola?

A: The value of $c$ determines the vertical shift of the parabola. If $c$ is positive, the parabola is shifted upwards. If $c$ is negative, the parabola is shifted downwards.

Q: How do I use a parabola to model real-world situations?

A: You can use a parabola to model various real-world situations, such as the trajectory of a projectile or the growth of a population. You can also use a parabola to solve problems involving quadratic equations.

Q: What are some common applications of parabolas?

A: Some common applications of parabolas include:

• Modeling the trajectory of a projectile
• Modeling the growth of a population
• Solving problems involving quadratic equations
• Designing optical systems, such as telescopes and microscopes
• Designing electrical circuits, such as filters and amplifiers

Q: What are some common mistakes to avoid when working with parabolas?

A: Some common mistakes to avoid when working with parabolas include:

• Not checking the sign of $a$ before solving the equation
• Not using the correct values of $a$, $b$, and $c$ in the equation
• Not checking the validity of the solution before presenting it
• Not considering the context of the problem when solving it

Q: Where can I find more information about parabolas?

A: You can find more information about parabolas in various resources, including:

• Textbooks on algebra and geometry
• Online resources, such as Khan Academy and Math Is Fun
• Research papers and articles on mathematics and science
• Online communities and forums, such as Reddit's r/learnmath and r/math