The Vertex Form and the Graph

Oct 13, 2025 by windailys.com 107 views

Introduction

In mathematics, the vertex form of a quadratic function is a powerful tool for understanding the graph of the function. The vertex form is given by $p(x)=a(x-h)^2+k$ , where $(h,k)$ is the vertex of the parabola. In this article, we will explore the function $p(x)=6x^2+48x$ written in vertex form as $p(x)=6(x+4)^2-96$ . We will examine the statements about the graph of $p$ and determine which ones are true.

The Axis of Symmetry

The axis of symmetry is the vertical line that passes through the vertex of the parabola. In the vertex form, the axis of symmetry is given by the equation $x=h$ . In this case, the vertex form is $p(x)=6(x+4)^2-96$ , so the axis of symmetry is the line $x=-4$ . Therefore, Statement 1: The axis of symmetry is the line $x=-4$ is TRUE.

The Vertex

The vertex of the parabola is the point $(h,k)$ . In the vertex form, the vertex is given by the point $(-4,-96)$ . Therefore, the vertex of the graph of $p$ is the point $(-4,-96)$ .

The Graph

The graph of a quadratic function in vertex form is a parabola that opens upward or downward. In this case, the graph of $p$ opens upward because the coefficient of the squared term is positive. The graph is also narrower than the graph of $y=x^2$ because the coefficient of the squared term is greater than 1.

The Vertex Form and the Graph

The vertex form of a quadratic function is a powerful tool for understanding the graph of the function. By writing the function in vertex form, we can easily identify the vertex of the parabola and the axis of symmetry. In this case, the vertex form is $p(x)=6(x+4)^2-96$ , so the vertex of the graph of $p$ is the point $(-4,-96)$ and the axis of symmetry is the line $x=-4$ .

Conclusion

In conclusion, the statements about the graph of $p$ are:

Statement 1: The axis of symmetry is the line $x=-4$ is TRUE.
Statement 2: The graph is narrower than the graph of $y=x^2$ is TRUE.
Statement 3: The graph is wider than the graph of $y=x^2$ is FALSE.

Therefore, the graph of $p$ is a parabola that opens upward and is narrower than the graph of $y=x^2$ .

Understanding the Vertex Form

The Vertex Form and the Axis of Symmetry

The vertex form of a quadratic function is given by $p(x)=a(x-h)^2+k$ , where $(h,k)$ is the vertex of the parabola. In this case, the vertex form is $p(x)=6(x+4)^2-96$ , so the axis of symmetry is the line $x=-4$ .

The Vertex Form and the Graph

Conclusion

In conclusion, the statements about the graph of $p$ are:

Statement 1: The axis of symmetry is the line $x=-4$ is TRUE.
Statement 2: The graph is narrower than the graph of $y=x^2$ is TRUE.
Statement 3: The graph is wider than the graph of $y=x^2$ is FALSE.

Therefore, the graph of $p$ is a parabola that opens upward and is narrower than the graph of $y=x^2$ .

The Vertex Form and the Standard Form

The vertex form of a quadratic function is given by $p(x)=a(x-h)^2+k$ , where $(h,k)$ is the vertex of the parabola. The standard form of a quadratic function is given by $p(x)=ax^2+bx+c$ . We can convert the standard form to the vertex form by completing the square.

Completing the Square

To complete the square, we need to add and subtract the square of half the coefficient of the linear term. In this case, the coefficient of the linear term is 48, so we need to add and subtract $(48/2)^2=144$ .

The Vertex Form

By completing the square, we can convert the standard form to the vertex form. In this case, the standard form is $p(x)=6x^2+48x$ , so the vertex form is $p(x)=6(x+4)^2-96$ .

The Axis of Symmetry

The Vertex

The vertex of the parabola is the point $(h,k)$ . In the vertex form, the vertex is given by the point $(-4,-96)$ . Therefore, the vertex of the graph of $p$ is the point $(-4,-96)$ .

The Graph

Conclusion

In conclusion, the statements about the graph of $p$ are:

Statement 1: The axis of symmetry is the line $x=-4$ is TRUE.
Statement 2: The graph is narrower than the graph of $y=x^2$ is TRUE.
Statement 3: The graph is wider than the graph of $y=x^2$ is FALSE.

Q: What is the vertex form of a quadratic function?

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

Q: How do I convert the standard form to the vertex form?

A: To convert the standard form to the vertex form, you need to complete the square. This involves adding and subtracting the square of half the coefficient of the linear term.

Q: What is the axis of symmetry in the vertex form?

A: The axis of symmetry in the vertex form is given by the equation $x=h$ . In the vertex form $p(x)=6(x+4)^2-96$ , the axis of symmetry is the line $x=-4$ .

Q: What is the vertex of the parabola in the vertex form?

A: The vertex of the parabola in the vertex form is given by the point $(h,k)$ . In the vertex form $p(x)=6(x+4)^2-96$ , the vertex is the point $(-4,-96)$ .

Q: How do I determine if the graph of a quadratic function opens upward or downward?

A: To determine if the graph of a quadratic function opens upward or downward, you need to look at the coefficient of the squared term. If the coefficient is positive, the graph opens upward. If the coefficient is negative, the graph opens downward.

Q: How do I compare the width of two parabolas?

A: To compare the width of two parabolas, you need to look at the coefficient of the squared term. If the coefficient is greater than 1, the parabola is narrower than the graph of $y=x^2$ . If the coefficient is less than 1, the parabola is wider than the graph of $y=x^2$ .

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

A: Some common mistakes to avoid when working with quadratic functions include:

Not completing the square correctly when converting the standard form to the vertex form.
Not identifying the axis of symmetry correctly.
Not determining the vertex of the parabola correctly.
Not comparing the width of two parabolas correctly.

Q: How can I practice working with quadratic functions?

A: You can practice working with quadratic functions by:

Completing the square to convert the standard form to the vertex form.
Identifying the axis of symmetry and the vertex of the parabola.
Comparing the width of two parabolas.
Graphing quadratic functions and identifying their characteristics.

Additional Resources

Conclusion

In conclusion, the vertex form of a quadratic function is a powerful tool for understanding the graph of the function. By completing the square and identifying the axis of symmetry and the vertex of the parabola, you can determine the characteristics of the graph. Remember to practice working with quadratic functions to become more confident and proficient in your understanding of these functions.