The Quadratic Function and Its Roots

Oct 13, 2025 by miodeya.com 91 views

Introduction

When analyzing the graph of a quadratic function, it's essential to consider various factors that can influence its shape and position. One such factor is the presence of a repeated root, which can significantly impact the graph's behavior. In this article, we will delve into the characteristics of the graph of the quadratic function $$y=5x^2+50x+125$$, with a repeated root of $$x=-5$$, and determine which of the given options accurately describes its appearance.

The Quadratic Function and Its Roots

The given quadratic function is in the form $$y=ax^2+bx+c$$, where $$a=5$$, $$b=50$$, and $$c=125$$. To understand the graph's behavior, we need to examine the roots of the function. The roots are the values of $$x$$ that make the function equal to zero. In this case, we are given that the function has a repeated root of $$x=-5$$. This means that the function can be factored as $$y=a(x+5)^2$$.

Factoring the Quadratic Function

To factor the quadratic function, we can use the given repeated root to rewrite the function in the form $$y=a(x+5)^2$$. Expanding this expression, we get $$y=5(x+5)^2$$. This form of the function reveals that the graph is a parabola that opens upward, since the coefficient of the squared term is positive.

The Vertex of the Parabola

The vertex of a parabola is the point at which the graph changes direction. In the case of a quadratic function in the form $$y=a(x-h)^2+k$$, the vertex is located at the point $$(h,k)$$. For the given function $$y=5(x+5)^2$$, the vertex is located at the point $$(-5,0)$$. Since the vertex is located to the left of the $$x$$-axis, the graph of the function has a vertex to the left of the $$x$$-axis.

The Graph's Behavior

Now that we have examined the vertex of the parabola, let's consider the graph's behavior. Since the coefficient of the squared term is positive, the graph opens upward. This means that the graph will increase as $$x$$ increases. Additionally, since the vertex is located to the left of the $$x$$-axis, the graph will be above the $$x$$-axis for all values of $$x$$.

Conclusion

Based on our analysis, we can conclude that the graph of the quadratic function $$y=5x^2+50x+125$$, with a repeated root of $$x=-5$$, has a vertex to the left of the $$x$$-axis. This is because the vertex is located at the point $$(-5,0)$$, and the graph opens upward. Therefore, option A is the correct answer.

Final Answer

The final answer is A. The graph has a vertex to the left of the $$x$$-axis.

Additional Information

The graph of the quadratic function $$y=5x^2+50x+125$$, with a repeated root of $$x=-5$$, is a parabola that opens upward.
The vertex of the parabola is located at the point $$(-5,0)$$.
The graph is above the $$x$$-axis for all values of $$x$$.

References

[1] "Quadratic Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/quadratic.html
[2] "Graphing Quadratic Functions" by Purplemath. Retrieved from https://www.purplemath.com/modules/graphquad.htm

Introduction

In our previous article, we explored the characteristics of the graph of the quadratic function $$y=5x^2+50x+125$$, with a repeated root of $$x=-5$$. We determined that the graph has a vertex to the left of the $$x$$-axis. In this article, we will address some frequently asked questions related to this topic.

Q: What is a repeated root in a quadratic function?

A: A repeated root in a quadratic function is a root that is repeated twice. In other words, if a quadratic function has a repeated root of $$x=r$$, it can be factored as $$y=a(x-r)^2$$.

Q: How do you determine the vertex of a parabola?

A: The vertex of a parabola is the point at which the graph changes direction. For a quadratic function in the form $$y=a(x-h)^2+k$$, the vertex is located at the point $$(h,k)$$. To determine the vertex, you can rewrite the function in this form and identify the values of $$h$$ and $$k$$.

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

A: The vertex of a parabola is significant because it represents the minimum or maximum value of the function. For a quadratic function that opens upward, the vertex represents the minimum value of the function. For a quadratic function that opens downward, the vertex represents the maximum value of the function.

Q: How do you determine the direction in which a parabola opens?

A: To determine the direction in which a parabola opens, you can examine the coefficient of the squared term. If the coefficient is positive, the parabola opens upward. If the coefficient is negative, the parabola opens downward.

Q: What is the relationship between the roots of a quadratic function and its graph?

A: The roots of a quadratic function are the values of $$x$$ that make the function equal to zero. The graph of a quadratic function passes through the points $$(r,0)$$ and $$(s,0)$$, where $$r$$ and $$s$$ are the roots of the function.

Q: How do you factor a quadratic function with a repeated root?

A: To factor a quadratic function with a repeated root, you can rewrite the function in the form $$y=a(x-r)^2$$. This form reveals that the graph is a parabola that opens upward or downward, depending on the sign of the coefficient $$a$$.

Q: What is the significance of the axis of symmetry in a quadratic function?

A: The axis of symmetry in a quadratic function is a vertical line that passes through the vertex of the parabola. The axis of symmetry is significant because it divides the graph into two symmetrical parts.

Q: How do you determine the axis of symmetry of a parabola?

A: To determine the axis of symmetry of a parabola, you can identify the vertex of the parabola and draw a vertical line through it. The equation of the axis of symmetry is $$x=h$$, where $$(h,k)$$ is the vertex of the parabola.

Conclusion

In this article, we addressed some frequently asked questions related to the quadratic function $$y=5x^2+50x+125$$, with a repeated root of $$x=-5$$. We hope that this Q&A article has provided you with a better understanding of the characteristics of this function and its graph.

Final Answer

The final answer is that the graph of the quadratic function $$y=5x^2+50x+125$$, with a repeated root of $$x=-5$$, has a vertex to the left of the $$x$$-axis.

Additional Information

The graph of the quadratic function $$y=5x^2+50x+125$$, with a repeated root of $$x=-5$$, is a parabola that opens upward.
The vertex of the parabola is located at the point $$(-5,0)$$.
The graph is above the $$x$$-axis for all values of $$x$$.

References

[1] "Quadratic Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/quadratic.html
[2] "Graphing Quadratic Functions" by Purplemath. Retrieved from https://www.purplemath.com/modules/graphquad.htm