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. { X $}$ Intercepts: { (-4,0), (-2,0)$}$ B.

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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 crosses 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 - 2x - 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 = -2.

x = -(-2)/(2*1)
x = 2/2
x = 1

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

y = (1)^2 - 2(1) - 8
y = 1 - 2 - 8
y = -9

Therefore, the vertex of the graph is (1, -9).

Finding the x-Intercepts

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

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

This equation has two solutions: x = 4 and x = -2. Therefore, the x-intercepts of the graph are (4, 0) and (-2, 0).

Conclusion

In conclusion, the vertex of the graph of the quadratic function y = (x-4)(x+2) is (1, -9), 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 crosses the x-axis.
  • Understanding the vertex and x-intercepts of a quadratic function is essential in mathematics.

Real-World Applications

The concept of vertex and x-intercepts has numerous real-world applications, including:

  • Physics: The vertex and x-intercepts of a quadratic function can be used to model the motion of an object under the influence of gravity.
  • Engineering: The vertex and x-intercepts of a quadratic function can be used to design and optimize the shape of a curve or a surface.
  • Economics: The vertex and x-intercepts of a quadratic function can be used to model the behavior of a market or an economy.

Final Thoughts

Frequently Asked Questions

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.

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.

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

A: The x-intercepts of a quadratic function are the points where the graph crosses 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.

Q: What is the difference between the vertex and x-intercepts?

A: The vertex is the maximum or minimum point of the parabola, while the x-intercepts are the points where the graph crosses the x-axis.

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

A: No, a quadratic function can have at most two x-intercepts.

Q: Can a quadratic function have no x-intercepts?

A: Yes, a quadratic function can have no x-intercepts if the graph does not cross the x-axis.

Q: How do I determine if a quadratic function has a maximum or minimum vertex?

A: To determine if a quadratic function has a maximum or minimum vertex, you can look at the coefficient of the x^2 term. If it is positive, the vertex is a minimum. If it is negative, the vertex is a maximum.

Q: Can a quadratic function have a vertex that is not on the x-axis?

A: Yes, a quadratic function can have a vertex that is not on the x-axis.

Q: How do I find the equation of a quadratic function given its vertex and x-intercepts?

A: To find the equation of a quadratic function given its vertex and x-intercepts, you can use the formula y = a(x - h)^2 + k, where (h, k) is the vertex and a is a constant.

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

A: No, a quadratic function can only have a vertex that is a maximum or minimum.

Q: How do I determine if a quadratic function is a function or not?

A: To determine if a quadratic function is a function or not, you can check if it passes the vertical line test. If it does, it is a function.

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

A: No, a quadratic function can only have a vertex that is a point.

Conclusion

In conclusion, the vertex and x-intercepts of a quadratic function are two essential concepts that help us understand the behavior of the function's graph. By understanding these concepts, we can analyze the behavior of the function and make predictions about its behavior. The concept of vertex and x-intercepts has numerous real-world applications, and it is essential in mathematics, physics, engineering, and economics.

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 crosses the x-axis.
  • Understanding the vertex and x-intercepts of a quadratic function is essential in mathematics.
  • The concept of vertex and x-intercepts has numerous real-world applications.