Forgetting to plug in values

Oct 13, 2025 by windailys.com 141 views

Understanding the Problem

The given equation $y=3x^2-6x+9$ represents a parabola, and we are tasked with finding the values of the high or low point of this parabola. To do this, we need to find the vertex of the parabola, which is the point where the parabola changes direction.

The Vertex Formula

The vertex of a parabola given by the equation $y=ax^2+bx+c$ can be found using the formula:

x=-\frac{b}{2a}

In our case, $a=3$ and $b=-6$ , so we can plug these values into the formula to find the x-coordinate of the vertex.

Finding the x-Coordinate of the Vertex

Using the formula, we get:

x=-\frac{-6}{2(3)}=\frac{6}{6}=1

So, the x-coordinate of the vertex is $x=1$ .

Finding the y-Coordinate of the Vertex

Now that we have the x-coordinate of the vertex, we can plug this value into the original equation to find the y-coordinate of the vertex.

y=3(1)^2-6(1)+9=3-6+9=6

So, the y-coordinate of the vertex is $y=6$ .

Completing the Table

Now that we have found the values of the vertex, we can complete the table:

x	y
-3	54
-2	33
-1	30
0	21
1	6

Discussion

The vertex of the parabola is the point where the parabola changes direction. In this case, the vertex is at the point $(1,6)$ . This means that the parabola opens upwards, and the vertex is the highest point on the parabola.

Conclusion

In this article, we have found the values of the high or low point of the parabola given by the equation $y=3x^2-6x+9$ . We used the vertex formula to find the x-coordinate of the vertex, and then plugged this value into the original equation to find the y-coordinate of the vertex. We also completed a table with the values of the vertex and other points on the parabola.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

Find the x-coordinate of the vertex: Use the formula $x=-\frac{b}{2a}$ to find the x-coordinate of the vertex.
Find the y-coordinate of the vertex: Plug the x-coordinate of the vertex into the original equation to find the y-coordinate of the vertex.
Complete the table: Use the values of the vertex and other points on the parabola to complete the table.

Example

Here is an example of how to use the vertex formula to find the x-coordinate of the vertex:

Suppose we have the equation $y=2x^2+4x+3$ . To find the x-coordinate of the vertex, we can use the formula:

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

So, the x-coordinate of the vertex is $x=-1$ .

Tips and Tricks

Here are some tips and tricks to help you solve this problem:

Use the vertex formula: The vertex formula is a quick and easy way to find the x-coordinate of the vertex.
Plug in values: Plug in values into the original equation to find the y-coordinate of the vertex.
Complete the table: Use the values of the vertex and other points on the parabola to complete the table.

Common Mistakes

Here are some common mistakes to avoid when solving this problem:

Forgetting to plug in values: Make sure to plug in values into the original equation to find the y-coordinate of the vertex.
Forgetting to complete the table: Make sure to complete the table with the values of the vertex and other points on the parabola.

Conclusion

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

Q: What is the vertex formula?

A: The vertex formula is a mathematical formula used to find the x-coordinate of the vertex of a parabola. It is given by: $x=-\frac{b}{2a}$ .

Q: How do I use the vertex formula?

A: To use the vertex formula, you need to identify the values of $a$ and $b$ in the equation of the parabola. Then, you can plug these values into the formula to find the x-coordinate of the vertex.

Q: What is the significance of the vertex of a parabola?

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

Q: Can I find the vertex of a parabola without using the vertex formula?

A: Yes, you can find the vertex of a parabola without using the vertex formula. You can complete the square to find the vertex, or you can use other methods such as graphing or using a calculator.

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

A: To complete the square, you need to rewrite the equation of the parabola in the form $y=a(x-h)^2+k$ . This will give you the vertex of the parabola, which is the point $(h,k)$ .

Q: What is the difference between the vertex and the axis of symmetry?

A: The vertex and the axis of symmetry are related but distinct concepts. The vertex is the point where the parabola changes direction, while the axis of symmetry is the vertical line that passes through the vertex and is perpendicular to the parabola.

Q: Can I find the axis of symmetry of a parabola without finding the vertex?

A: Yes, you can find the axis of symmetry of a parabola without finding the vertex. You can use the formula $x=-\frac{b}{2a}$ to find the x-coordinate of the axis of symmetry, or you can use other methods such as graphing or using a calculator.

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

A: To use the axis of symmetry to find the vertex of a parabola, you need to find the x-coordinate of the axis of symmetry and then plug this value into the original equation to find the y-coordinate of the vertex.

Q: What is the relationship between the vertex and the roots of a parabola?

A: The vertex and the roots of a parabola are related in that the vertex is the midpoint of the segment connecting the roots. This means that the x-coordinate of the vertex is the average of the x-coordinates of the roots.

Q: Can I find the roots of a parabola without finding the vertex?

A: Yes, you can find the roots of a parabola without finding the vertex. You can use the quadratic formula to find the roots, or you can use other methods such as graphing or using a calculator.

Q: How do I use the quadratic formula to find the roots of a parabola?

A: To use the quadratic formula to find the roots of a parabola, you need to identify the values of $a$ , $b$ , and $c$ in the equation of the parabola. Then, you can plug these values into the quadratic formula to find the roots.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula used to find the roots of a quadratic equation. It is given by: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ .

Q: How do I use the quadratic formula to find the roots of a parabola?

Q: Can I find the vertex of a parabola without using the quadratic formula?

A: Yes, you can find the vertex of a parabola without using the quadratic formula. You can use other methods such as graphing or using a calculator.

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

A: To use a calculator to find the vertex of a parabola, you need to enter the equation of the parabola into the calculator and then use the calculator's built-in functions to find the vertex.

Q: What are some common mistakes to avoid when finding the vertex of a parabola?

A: Some common mistakes to avoid when finding the vertex of a parabola include forgetting to plug in values, forgetting to complete the square, and using the wrong formula.

Q: How do I check my work when finding the vertex of a parabola?

A: To check your work when finding the vertex of a parabola, you can use the following steps:

Plug in values: Plug in values into the original equation to find the y-coordinate of the vertex.
Check the formula: Check that you are using the correct formula to find the vertex.
Check the calculations: Check that your calculations are correct.
Check the graph: Check that the graph of the parabola is consistent with the vertex you found.

Q: What are some real-world applications of finding the vertex of a parabola?

A: Some real-world applications of finding the vertex of a parabola include:

Physics: Finding the vertex of a parabola can be used to model the motion of objects under the influence of gravity.
Engineering: Finding the vertex of a parabola can be used to design curves and surfaces for buildings and other structures.
Computer Science: Finding the vertex of a parabola can be used to optimize algorithms and data structures.

Q: Can I find the vertex of a parabola using a graphing calculator?

A: Yes, you can find the vertex of a parabola using a graphing calculator. Simply enter the equation of the parabola into the calculator and then use the calculator's built-in functions to find the vertex.

Q: How do I use a graphing calculator to find the vertex of a parabola?

A: To use a graphing calculator to find the vertex of a parabola, you need to enter the equation of the parabola into the calculator and then use the calculator's built-in functions to find the vertex.

Q: What are some common mistakes to avoid when using a graphing calculator to find the vertex of a parabola?

A: Some common mistakes to avoid when using a graphing calculator to find the vertex of a parabola include:

Forgetting to enter the equation correctly: Make sure to enter the equation of the parabola correctly into the calculator.
Forgetting to use the correct function: Make sure to use the correct function to find the vertex.
Forgetting to check the calculations: Make sure to check that your calculations are correct.

Q: How do I check my work when using a graphing calculator to find the vertex of a parabola?

A: To check your work when using a graphing calculator to find the vertex of a parabola, you can use the following steps:

Check the equation: Check that you entered the equation of the parabola correctly into the calculator.
Check the function: Check that you used the correct function to find the vertex.
Check the calculations: Check that your calculations are correct.
Check the graph: Check that the graph of the parabola is consistent with the vertex you found.

Q: What are some real-world applications of using a graphing calculator to find the vertex of a parabola?

A: Some real-world applications of using a graphing calculator to find the vertex of a parabola include:

Physics: Using a graphing calculator to find the vertex of a parabola can be used to model the motion of objects under the influence of gravity.
Engineering: Using a graphing calculator to find the vertex of a parabola can be used to design curves and surfaces for buildings and other structures.
Computer Science: Using a graphing calculator to find the vertex of a parabola can be used to optimize algorithms and data structures.