Complete The Following Table Using The Equation: Y = X 2 Y = X^2 Y = X 2 ${ \begin{array}{|c|c|c|c|c|c|} \hline x & -3 & -1 & 0 & 1 & 3 \ \hline y & & & & & \ \hline \end{array} }$Plot The Points On A Graph And Determine Which Graph Best

In this article, we will explore the equation $y = x^2$ and use it to complete a given table. We will then plot the points on a graph and determine which graph best represents the equation.

The Equation $y = x^2$

The equation $y = x^2$ is a quadratic equation that represents a parabola. The graph of this equation is a U-shaped curve that opens upwards. The equation can be rewritten as $y = x \cdot x$, which shows that the value of $y$ is equal to the product of $x$ and itself.

Completing the Table

To complete the table, we need to substitute the values of $x$ into the equation $y = x^2$ and calculate the corresponding values of $y$.

x	y
-3
-1
0
1
3

Let's start by substituting the value of $x = -3$ into the equation:

$$y = (-3)^2$$

Using the order of operations, we first evaluate the exponent:

$$(-3)^2 = -3 \cdot -3 = 9$$

So, the value of $y$ when $x = -3$ is 9.

x	y
-3	9
-1
0
1
3

Next, let's substitute the value of $x = -1$ into the equation:

$$y = (-1)^2$$

Using the order of operations, we first evaluate the exponent:

$$(-1)^2 = -1 \cdot -1 = 1$$

So, the value of $y$ when $x = -1$ is 1.

x	y
-3	9
-1	1
0
1
3

Now, let's substitute the value of $x = 0$ into the equation:

$$y = (0)^2$$

Using the order of operations, we first evaluate the exponent:

$$(0)^2 = 0 \cdot 0 = 0$$

So, the value of $y$ when $x = 0$ is 0.

x	y
-3	9
-1	1
0	0
1
3

Next, let's substitute the value of $x = 1$ into the equation:

$$y = (1)^2$$

Using the order of operations, we first evaluate the exponent:

$$(1)^2 = 1 \cdot 1 = 1$$

So, the value of $y$ when $x = 1$ is 1.

x	y
-3	9
-1	1
0	0
1	1
3

Finally, let's substitute the value of $x = 3$ into the equation:

$$y = (3)^2$$

Using the order of operations, we first evaluate the exponent:

$$(3)^2 = 3 \cdot 3 = 9$$

So, the value of $y$ when $x = 3$ is 9.

x	y
-3	9
-1	1
0	0
1	1
3	9

Plotting the Points on a Graph

To plot the points on a graph, we need to use a coordinate plane. The x-axis represents the values of $x$, and the y-axis represents the values of $y$.

x	y
-3	9
-1	1
0	0
1	1
3	9

Plotting the points on the graph, we get the following:

• The point (-3, 9) is plotted on the graph.
• The point (-1, 1) is plotted on the graph.
• The point (0, 0) is plotted on the graph.
• The point (1, 1) is plotted on the graph.
• The point (3, 9) is plotted on the graph.

Determining the Graph

The graph of the equation $y = x^2$ is a parabola that opens upwards. The points plotted on the graph are all on the parabola, which means that the graph is a perfect representation of the equation.

Conclusion

In this article, we completed a table using the equation $y = x^2$ and plotted the points on a graph. We determined that the graph is a parabola that opens upwards and is a perfect representation of the equation.

Key Takeaways

• The equation $y = x^2$ represents a parabola that opens upwards.
• The graph of the equation is a perfect representation of the equation.
• The points plotted on the graph are all on the parabola.

Final Thoughts

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In this article, we will answer some frequently asked questions about the equation $y = x^2$. We will cover topics such as the graph of the equation, the properties of the parabola, and how to use the equation in real-world applications.

Q: What is the graph of the equation $y = x^2$?

A: The graph of the equation $y = x^2$ is a parabola that opens upwards. The parabola is a U-shaped curve that is symmetric about the y-axis.

Q: What are the properties of the parabola?

A: The parabola has several properties, including:

• Symmetry: The parabola is symmetric about the y-axis.
• Vertex: The vertex of the parabola is at the point (0, 0).
• Axis of symmetry: The axis of symmetry of the parabola is the y-axis.
• Opening direction: The parabola opens upwards.

Q: How do I use the equation $y = x^2$ in real-world applications?

A: The equation $y = x^2$ can be used to model many real-world phenomena, such as:

• Projectile motion: The equation can be used to model the trajectory of a projectile, such as a thrown ball or a rocket.
• Optics: The equation can be used to model the behavior of light as it passes through a lens or a mirror.
• Physics: The equation can be used to model the motion of an object under the influence of gravity.

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

A: To find the vertex of the parabola, you can use the formula:

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

where $a$ and $b$ are the coefficients of the quadratic equation.

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

A: To find the axis of symmetry of the parabola, you can use the formula:

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

where $a$ and $b$ are the coefficients of the quadratic equation.

Q: How do I graph the equation $y = x^2$?

A: To graph the equation $y = x^2$, you can use a graphing calculator or a computer program. You can also plot the points on a graph and connect them to form the parabola.

Q: What are some common mistakes to avoid when working with the equation $y = x^2$?

A: Some common mistakes to avoid when working with the equation $y = x^2$ include:

• Not using the correct formula: Make sure to use the correct formula for the equation, which is $y = x^2$.
• Not considering the domain: Make sure to consider the domain of the equation, which is all real numbers.
• Not considering the range: Make sure to consider the range of the equation, which is all non-negative real numbers.

Conclusion

In this article, we answered some frequently asked questions about the equation $y = x^2$. We covered topics such as the graph of the equation, the properties of the parabola, and how to use the equation in real-world applications. We also discussed some common mistakes to avoid when working with the equation.

Key Takeaways

• The graph of the equation $y = x^2$ is a parabola that opens upwards.
• The parabola has several properties, including symmetry, vertex, axis of symmetry, and opening direction.
• The equation can be used to model many real-world phenomena, such as projectile motion, optics, and physics.
• To find the vertex of the parabola, you can use the formula $x = -\frac{b}{2a}$.
• To find the axis of symmetry of the parabola, you can use the formula $x = -\frac{b}{2a}$.
• To graph the equation $y = x^2$, you can use a graphing calculator or a computer program.

Final Thoughts

The equation $y = x^2$ is a simple yet powerful equation that can be used to model many real-world phenomena. By understanding the graph of the equation, the properties of the parabola, and how to use the equation in real-world applications, you can gain a deeper understanding of the equation and its uses.