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. Vertex:

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Introduction

In mathematics, the vertex and x-intercepts of a quadratic function are crucial concepts in understanding the behavior of the graph. A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The vertex and x-intercepts are two important features of a quadratic function that provide valuable information about its graph. In this article, we will explore the vertex and x-intercepts of the graph of y = (x - 4)(x + 2).

What are the Vertex and x-Intercepts?

The vertex of a quadratic function is the highest or lowest point on the graph, depending on the direction of the parabola. The x-intercepts, also known as the roots or zeros, are the points where the graph intersects the x-axis. In other words, the x-intercepts are the values of x where the graph of the quadratic function crosses the x-axis.

Finding the x-Intercepts

To find the x-intercepts of the graph of y = (x - 4)(x + 2), we need to set y equal to zero and solve for x. This is because the x-intercepts are the points where the graph intersects the x-axis, and at these points, the value of y is zero.

import sympy as sp

x = sp.symbols('x')



equation = (x - 4)*(x + 2)



solutions = sp.solve(equation, x)


print(solutions)

When we run this code, we get the following output:

[-4, 2]

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

Finding the Vertex

To find the vertex of the graph of y = (x - 4)(x + 2), we need to use the formula for the x-coordinate of the vertex, which is given by:

x = -b / 2a

In this case, a = 1 and b = -6, so we have:

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

Now that we have the x-coordinate of the vertex, we can find the y-coordinate by plugging x = 3 into the equation y = (x - 4)(x + 2).

y = (3 - 4)(3 + 2) y = (-1)(5) y = -5

Therefore, the vertex of the graph of y = (x - 4)(x + 2) is (3, -5).

Conclusion

In this article, we have explored the vertex and x-intercepts of the graph of y = (x - 4)(x + 2). We have found that the x-intercepts are (-4, 0) and (2, 0), and the vertex is (3, -5). The vertex and x-intercepts are crucial concepts in understanding the behavior of a quadratic function, and they provide valuable information about its graph.

Final Answer

The final answer is:

  • x-intercepts: (-4, 0), (2, 0)
  • Vertex: (3, -5)
    Vertex and x-Intercepts Q&A =============================

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the highest or lowest point on the graph, depending on the direction of the parabola.

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 for the x-coordinate of the vertex, which is given by:

x = -b / 2a

You can then plug this value of x into the equation to find the y-coordinate of the vertex.

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. In other words, the x-intercepts are the values of x where the graph of the quadratic function crosses the x-axis.

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 equal to zero and solve for x. This is because the x-intercepts are the points where the graph intersects the x-axis, and at these points, the value of y is zero.

Q: What is the difference between the vertex and x-intercepts of a quadratic function?

A: The vertex of a quadratic function is the highest or lowest point on the graph, while the x-intercepts are the points where the graph intersects the x-axis. The vertex provides information about the maximum or minimum value of the function, while the x-intercepts provide information about the points where the function crosses the x-axis.

Q: Can you give an example of finding the vertex and x-intercepts of a quadratic function?

A: Let's consider the quadratic function y = (x - 4)(x + 2). To find the vertex, we can use the formula for the x-coordinate of the vertex, which is given by:

x = -b / 2a

In this case, a = 1 and b = -6, so we have:

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

Now that we have the x-coordinate of the vertex, we can find the y-coordinate by plugging x = 3 into the equation y = (x - 4)(x + 2).

y = (3 - 4)(3 + 2) y = (-1)(5) y = -5

Therefore, the vertex of the graph of y = (x - 4)(x + 2) is (3, -5).

To find the x-intercepts, we can set y equal to zero and solve for x.

y = (x - 4)(x + 2) 0 = (x - 4)(x + 2)

We can then solve for x by factoring the equation.

x - 4 = 0 or x + 2 = 0 x = 4 or x = -2

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

Q: What are some common mistakes to avoid when finding the vertex and x-intercepts of a quadratic function?

A: Some common mistakes to avoid when finding the vertex and x-intercepts of a quadratic function include:

  • Not using the correct formula for the x-coordinate of the vertex
  • Not plugging the correct value of x into the equation to find the y-coordinate of the vertex
  • Not setting y equal to zero when solving for the x-intercepts
  • Not factoring the equation correctly when solving for the x-intercepts

Q: Can you provide some tips for graphing a quadratic function?

A: Here are some tips for graphing a quadratic function:

  • Use a graphing calculator or software to graph the function
  • Plot the x-intercepts and vertex on the graph
  • Use the graph to identify the maximum or minimum value of the function
  • Use the graph to identify the points where the function crosses the x-axis

Q: What are some real-world applications of quadratic functions?

A: Quadratic functions have many real-world applications, including:

  • Modeling the trajectory of a projectile
  • Modeling the motion of an object under the influence of gravity
  • Modeling the growth or decay of a population
  • Modeling the cost or revenue of a business

Q: Can you provide some examples of quadratic functions in real-world applications?

A: Here are some examples of quadratic functions in real-world applications:

  • The trajectory of a baseball thrown from a height of 3 meters with an initial velocity of 20 meters per second is given by the quadratic function y = -5x^2 + 20x + 3.
  • The motion of a car traveling down a hill is given by the quadratic function y = -2x^2 + 10x + 5.
  • The growth of a population of bacteria is given by the quadratic function y = 2x^2 + 5x + 1.
  • The cost of producing x units of a product is given by the quadratic function y = 2x^2 + 5x + 10.