What Are The Vertex And $x$-intercepts Of The Graph Of $y=(x+4)(x+2)$?Select One Answer For The Vertex And One For The $x$-intercepts.A. Vertex: $(-3,-1)$ B. $x$-Intercepts:

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Introduction

In mathematics, the vertex and x-intercepts of a quadratic function are two essential concepts that help us understand the behavior of the function's graph. The vertex represents the maximum or minimum point of the parabola, while the x-intercepts are the points where the graph intersects the x-axis. In this article, we will explore the vertex and x-intercepts of the graph of the quadratic function y = (x + 4)(x + 2).

The Quadratic Function

The given quadratic function is y = (x + 4)(x + 2). To find the vertex and x-intercepts, we need to expand the function and rewrite it in the standard form of a quadratic function, which is y = ax^2 + bx + c.

y = (x + 4)(x + 2)
y = x^2 + 2x + 4x + 8
y = x^2 + 6x + 8

Finding the Vertex

The vertex of a quadratic function can be found using the formula x = -b / 2a, where a and b are the coefficients of the quadratic function. In this case, a = 1 and b = 6.

x = -b / 2a
x = -6 / 2(1)
x = -6 / 2
x = -3

To find the y-coordinate of the vertex, we substitute the x-coordinate into the quadratic function.

y = x^2 + 6x + 8
y = (-3)^2 + 6(-3) + 8
y = 9 - 18 + 8
y = -1

Therefore, the vertex of the graph of y = (x + 4)(x + 2) is (-3, -1).

Finding the x-Intercepts

The x-intercepts of a quadratic function are the points where the graph intersects the x-axis. To find the x-intercepts, we set y = 0 and solve for x.

0 = x^2 + 6x + 8
0 = (x + 4)(x + 2)

This equation can be factored into two linear equations, which are x + 4 = 0 and x + 2 = 0.

x + 4 = 0
x = -4

x + 2 = 0
x = -2

Therefore, the x-intercepts of the graph of y = (x + 4)(x + 2) are (-4, 0) and (-2, 0).

Conclusion

In conclusion, the vertex of the graph of y = (x + 4)(x + 2) is (-3, -1), and the x-intercepts are (-4, 0) and (-2, 0). Understanding the vertex and x-intercepts of a quadratic function is essential in mathematics, as it helps us analyze the behavior of the function's graph and make predictions about its behavior.

Key Takeaways

  • The vertex of a quadratic function can be found using the formula x = -b / 2a.
  • The x-intercepts of a quadratic function are the points where the graph intersects the x-axis.
  • The vertex and x-intercepts of a quadratic function can be used to analyze the behavior of the function's graph.

Final Answer

Based on the calculations above, the final answer is:

  • Vertex: (-3, -1)
  • x-Intercepts: (-4, 0) and (-2, 0)
    Vertex and x-Intercepts of a Quadratic Function: Q&A =====================================================

Introduction

In our previous article, we explored the vertex and x-intercepts of the graph of the quadratic function y = (x + 4)(x + 2). In this article, we will answer some frequently asked questions about the vertex and x-intercepts of a quadratic function.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the maximum or minimum point of the parabola. It is the point where the graph changes direction, either from increasing to decreasing or from decreasing to increasing.

Q: How do I find the vertex of a quadratic function?

A: To find the vertex of a quadratic function, you can use the formula x = -b / 2a, where a and b are the coefficients of the quadratic function. You can then substitute the x-coordinate into the quadratic function to find the y-coordinate.

Q: What are the x-intercepts of a quadratic function?

A: The x-intercepts of a quadratic function are the points where the graph intersects the x-axis. They are the solutions to the equation y = 0.

Q: How do I find the x-intercepts of a quadratic function?

A: To find the x-intercepts of a quadratic function, you can set y = 0 and solve for x. You can use factoring, the quadratic formula, or other methods to find the solutions.

Q: Can a quadratic function have more than two x-intercepts?

A: No, a quadratic function can have at most two x-intercepts. This is because a quadratic function is a polynomial of degree 2, and it can intersect the x-axis at most twice.

Q: Can a quadratic function have a vertex that is not an x-intercept?

A: Yes, a quadratic function can have a vertex that is not an x-intercept. This occurs when the vertex is located between the two x-intercepts.

Q: How do I determine the direction of the parabola?

A: To determine the direction of the parabola, you can look at the coefficient of the x^2 term. If the coefficient is positive, the parabola opens upward, and if it is negative, the parabola opens downward.

Q: Can a quadratic function have a vertex that is not a maximum or minimum point?

A: No, a quadratic function can only have a vertex that is a maximum or minimum point. This is because the vertex is the point where the graph changes direction, and it is the only point where the graph can change direction.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you can use the vertex and x-intercepts to plot points on the graph. You can then connect the points to form the parabola.

Conclusion

In conclusion, the vertex and x-intercepts of a quadratic function are essential concepts in mathematics. Understanding these concepts can help you analyze the behavior of the function's graph and make predictions about its behavior. We hope this Q&A article has helped you understand the vertex and x-intercepts of a quadratic function better.

Key Takeaways

  • The vertex of a quadratic function is the maximum or minimum point of the parabola.
  • The x-intercepts of a quadratic function are the points where the graph intersects the x-axis.
  • A quadratic function can have at most two x-intercepts.
  • A quadratic function can have a vertex that is not an x-intercept.
  • The direction of the parabola can be determined by the coefficient of the x^2 term.

Final Answer

Based on the questions and answers above, the final answer is:

  • The vertex of a quadratic function is the maximum or minimum point of the parabola.
  • The x-intercepts of a quadratic function are the points where the graph intersects the x-axis.
  • A quadratic function can have at most two x-intercepts.
  • A quadratic function can have a vertex that is not an x-intercept.
  • The direction of the parabola can be determined by the coefficient of the x^2 term.