Plotting Points on the Parabola

Oct 13, 2025 by windailys.com 7 views

Introduction

In mathematics, a parabola is a type of quadratic function that can be represented in the form $y=ax^2+bx+c$ , where $a$ , $b$ , and $c$ are constants. The parabola $y=2x^2-12x+14$ is a specific example of a quadratic function that we will graph and analyze in this article. We will first find the vertex of the parabola, and then plot five points on the parabola: the vertex, two points to the left of the vertex, and two points to the right of the vertex.

Finding the Vertex

The vertex of a parabola in the form $y=ax^2+bx+c$ can be found using the formula $x=-\frac{b}{2a}$ . In this case, $a=2$ and $b=-12$ , so we can plug these values into the formula to find the x-coordinate of the vertex:

$x=-\frac{-12}{2(2)}$ $x=-\frac{-12}{4}$ $x=\frac{12}{4}$ $x=3$

Now that we have the x-coordinate of the vertex, we can find the y-coordinate by plugging $x=3$ into the equation of the parabola:

$y=2(3)^2-12(3)+14$ $y=2(9)-36+14$ $y=18-36+14$ $y=-4$

So, the vertex of the parabola $y=2x^2-12x+14$ is at the point $(3, -4)$ .

Plotting Points on the Parabola

Now that we have the vertex of the parabola, we can plot five points on the parabola: the vertex, two points to the left of the vertex, and two points to the right of the vertex. We will use the equation of the parabola to find the y-coordinates of these points.

Points to the Left of the Vertex

Point 1: $(2, y)$

To find the y-coordinate of this point, we can plug $x=2$ into the equation of the parabola:

$y=2(2)^2-12(2)+14$ $y=2(4)-24+14$ $y=8-24+14$ $y=-2$

So, the point $(2, -2)$ is on the parabola.

Point 2: $(1, y)$

To find the y-coordinate of this point, we can plug $x=1$ into the equation of the parabola:

$y=2(1)^2-12(1)+14$ $y=2(1)-12+14$ $y=2-12+14$ $y=4$

So, the point $(1, 4)$ is on the parabola.

Points to the Right of the Vertex

Point 3: $(4, y)$

To find the y-coordinate of this point, we can plug $x=4$ into the equation of the parabola:

$y=2(4)^2-12(4)+14$ $y=2(16)-48+14$ $y=32-48+14$ $y=-2$

So, the point $(4, -2)$ is on the parabola.

Point 4: $(5, y)$

To find the y-coordinate of this point, we can plug $x=5$ into the equation of the parabola:

$y=2(5)^2-12(5)+14$ $y=2(25)-60+14$ $y=50-60+14$ $y=4$

So, the point $(5, 4)$ is on the parabola.

Plotting the Points

Now that we have the five points on the parabola, we can plot them on a coordinate plane. The points are:

$(3, -4)$ (vertex)
$(2, -2)$ (point to the left of the vertex)
$(1, 4)$ (point to the left of the vertex)
$(4, -2)$ (point to the right of the vertex)
$(5, 4)$ (point to the right of the vertex)

We can plot these points on a coordinate plane and draw a smooth curve through them to represent the parabola.

Conclusion

Q: What is the vertex of the parabola $y=2x^2-12x+14$ ?

A: The vertex of the parabola $y=2x^2-12x+14$ is at the point $(3, -4)$ . This is the lowest point on the parabola.

Q: How do I find the x-coordinate of the vertex?

A: To find the x-coordinate of the vertex, you can use the formula $x=-\frac{b}{2a}$ . In this case, $a=2$ and $b=-12$ , so you can plug these values into the formula to find the x-coordinate of the vertex.

Q: How do I find the y-coordinate of the vertex?

A: To find the y-coordinate of the vertex, you can plug the x-coordinate of the vertex into the equation of the parabola. In this case, the x-coordinate of the vertex is $x=3$ , so you can plug this value into the equation of the parabola to find the y-coordinate of the vertex.

Q: What are the points to the left of the vertex?

A: The points to the left of the vertex are $(2, -2)$ and $(1, 4)$ . These points are on the parabola and are to the left of the vertex.

Q: What are the points to the right of the vertex?

A: The points to the right of the vertex are $(4, -2)$ and $(5, 4)$ . These points are on the parabola and are to the right of the vertex.

Q: How do I plot the points on a coordinate plane?

A: To plot the points on a coordinate plane, you can use a ruler or a straightedge to draw a line through each point. You can also use a graphing calculator or a computer program to plot the points.

Q: What is the shape of the parabola?

A: The parabola $y=2x^2-12x+14$ is a quadratic function that opens upward. This means that the parabola has a minimum point, which is the vertex, and the parabola opens upward from this point.

Q: What is the equation of the parabola?

A: The equation of the parabola is $y=2x^2-12x+14$ . This is a quadratic function that can be represented in the form $y=ax^2+bx+c$ , where $a=2$ , $b=-12$ , and $c=14$ .

Q: How do I graph the parabola?

A: To graph the parabola, you can use a graphing calculator or a computer program to plot the points on the parabola. You can also use a ruler or a straightedge to draw a smooth curve through the points.

Q: What are some real-world applications of graphing parabolas?

A: Graphing parabolas has many real-world applications, including:

Modeling the trajectory of a projectile
Finding the maximum or minimum value of a function
Determining the shape of a curve
Analyzing the behavior of a system

These are just a few examples of the many real-world applications of graphing parabolas.