Vertex Form of the Quadratic Equation

Oct 13, 2025 by pynkocean.com 115 views

Introduction

The quadratic equation is a fundamental concept in mathematics, and it has numerous applications in various fields, including physics, engineering, and economics. In its standard form, the quadratic equation is represented as $y = ax^2 + bx + c$ . However, there is another form of the quadratic equation, known as the vertex form, which is represented as $y = a(x - h)^2 + k$ . In this article, we will focus on rewriting the quadratic equation in the form $y = a(x - h)^2 + k$ .

Given Quadratic Equation

The given quadratic equation is $y = 5x^2 - 30x + 95$ . To rewrite this equation in the vertex form, we need to complete the square.

Step 1: Factor Out the Coefficient of $x^2$

The first step in rewriting the quadratic equation is to factor out the coefficient of $x^2$ , which is 5.

y = 5(x^2 - 6x) + 95

Step 2: Complete the Square

To complete the square, we need to add and subtract $(\frac{b}{2})^2$ inside the parentheses, where $b$ is the coefficient of $x$ . In this case, $b = -6$ .

y = 5(x^2 - 6x + 9 - 9) + 95

Step 3: Simplify the Expression

Now, we can simplify the expression by combining like terms.

y = 5((x - 3)^2 - 9) + 95

Step 4: Distribute the Coefficient

Next, we need to distribute the coefficient 5 to the terms inside the parentheses.

y = 5(x - 3)^2 - 45 + 95

Step 5: Combine Like Terms

Finally, we can combine the like terms to get the final expression.

y = 5(x - 3)^2 + 50

Conclusion

In this article, we have successfully rewritten the quadratic equation $y = 5x^2 - 30x + 95$ in the form $y = a(x - h)^2 + k$ . The vertex form of the quadratic equation is a useful tool for graphing and analyzing quadratic functions. By completing the square, we can rewrite the quadratic equation in a form that reveals its vertex and axis of symmetry.

Vertex Form of the Quadratic Equation

The vertex form of the quadratic equation is $y = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola. In the rewritten equation $y = 5(x - 3)^2 + 50$ , we can see that the vertex is at the point $(3, 50)$ .

Graphing the Quadratic Equation

To graph the quadratic equation, we can use the vertex form. Since the vertex is at the point $(3, 50)$ , we can start by plotting the point $(3, 50)$ on the coordinate plane. Then, we can use the fact that the parabola is symmetric about the vertical line $x = h$ to plot the remaining points.

Applications of the Vertex Form

The vertex form of the quadratic equation has numerous applications in various fields, including physics, engineering, and economics. For example, in physics, the vertex form can be used to model the motion of an object under the influence of gravity. In engineering, the vertex form can be used to design and optimize systems that involve quadratic functions. In economics, the vertex form can be used to model the behavior of economic systems that involve quadratic functions.

Conclusion

Introduction

In our previous article, we discussed how to rewrite the quadratic equation in the form $y = a(x - h)^2 + k$ . In this article, we will answer some frequently asked questions about the quadratic equation in vertex form.

Q: What is the vertex form of the quadratic equation?

A: The vertex form of the quadratic equation is $y = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola.

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

A: To find the vertex of the parabola, you need to complete the square and rewrite the quadratic equation in the form $y = a(x - h)^2 + k$ . The vertex is then given by the point $(h, k)$ .

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

A: The vertex of the parabola is the highest or lowest point on the graph, depending on the value of $a$ . It is also the point of symmetry of the parabola.

Q: How do I graph the quadratic equation in vertex form?

A: To graph the quadratic equation in vertex form, you can start by plotting the vertex $(h, k)$ . Then, you can use the fact that the parabola is symmetric about the vertical line $x = h$ to plot the remaining points.

Q: Can I use the vertex form to solve quadratic equations?

A: Yes, you can use the vertex form to solve quadratic equations. By setting $y = 0$ and solving for $x$ , you can find the x-intercepts of the parabola.

Q: What are some real-world applications of the vertex form?

A: The vertex form has numerous real-world applications, including:

Modeling the motion of an object under the influence of gravity
Designing and optimizing systems that involve quadratic functions
Modeling the behavior of economic systems that involve quadratic functions

Q: Can I use the vertex form to find the maximum or minimum value of a quadratic function?

A: Yes, you can use the vertex form to find the maximum or minimum value of a quadratic function. The maximum or minimum value is given by the value of $y$ at the vertex $(h, k)$ .

Q: How do I determine whether the parabola opens upward or downward?

A: To determine whether the parabola opens upward or downward, you need to look at the value of $a$ . If $a > 0$ , the parabola opens upward. If $a < 0$ , the parabola opens downward.

Q: Can I use the vertex form to find the axis of symmetry of a parabola?

A: Yes, you can use the vertex form to find the axis of symmetry of a parabola. The axis of symmetry is given by the vertical line $x = h$ .

Conclusion

In conclusion, the vertex form of the quadratic equation is a powerful tool for graphing and analyzing quadratic functions. By completing the square, we can rewrite the quadratic equation in a form that reveals its vertex and axis of symmetry. The vertex form has numerous real-world applications, and it is an essential tool for anyone who works with quadratic functions.

Additional Resources

Practice Problems

Rewrite the quadratic equation $y = 2x^2 + 12x + 20$ in vertex form.
Graph the quadratic equation $y = -3(x - 2)^2 + 5$ .
Find the x-intercepts of the parabola given by the equation $y = x^2 - 6x + 8$ .
Determine whether the parabola given by the equation $y = 2(x - 1)^2 + 3$ opens upward or downward.
Find the axis of symmetry of the parabola given by the equation $y = -2(x + 2)^2 + 1$ .