Find The { X $} − I N T E R C E P T S ( I F A N Y ) F O R T H E G R A P H O F T H E Q U A D R A T I C F U N C T I O N . -intercepts (if Any) For The Graph Of The Quadratic Function. − In T Erce Pt S ( I F An Y ) F Or T H E G R A P H O F T H E Q U A D R A T I C F U N C T I O N . { F(x) = (x+3)^2 - 9 \} A. { (0,0) $}$ And { (-6,0) $}$B. { (6,0) $}$ And { (-6,0) $} C . \[ C. \[ C . \[ (0,9)

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Understanding the Problem

In this problem, we are given a quadratic function in the form of f(x)=(x+3)2−9f(x) = (x+3)^2 - 9. Our goal is to find the x-intercepts of the graph of this function, if any. The x-intercepts are the points where the graph of the function crosses the x-axis, meaning the y-coordinate of these points is 0.

What are x-intercepts?

The x-intercepts of a quadratic function are the solutions to the equation f(x)=0f(x) = 0. In other words, we need to find the values of x that make the function equal to 0. These values are also known as the roots or zeros of the function.

How to find x-intercepts?

To find the x-intercepts of a quadratic function, we can use the following steps:

  1. Set the function equal to 0: We start by setting the function equal to 0, which gives us the equation f(x)=0f(x) = 0.
  2. Simplify the equation: We then simplify the equation by expanding the squared term and combining like terms.
  3. Solve for x: Finally, we solve for x by isolating the variable x on one side of the equation.

Finding the x-intercepts of the given function

Now that we have understood the concept of x-intercepts and the steps to find them, let's apply these steps to the given function f(x)=(x+3)2−9f(x) = (x+3)^2 - 9.

Step 1: Set the function equal to 0

We start by setting the function equal to 0:

f(x)=0f(x) = 0

Step 2: Simplify the equation

We then simplify the equation by expanding the squared term and combining like terms:

(x+3)2−9=0(x+3)^2 - 9 = 0

Expanding the squared term, we get:

x2+6x+9−9=0x^2 + 6x + 9 - 9 = 0

Combining like terms, we get:

x2+6x=0x^2 + 6x = 0

Step 3: Solve for x

Finally, we solve for x by isolating the variable x on one side of the equation:

x2+6x=0x^2 + 6x = 0

We can factor out an x from the left-hand side of the equation:

x(x+6)=0x(x + 6) = 0

This gives us two possible solutions:

x=0x = 0 or x+6=0x + 6 = 0

Solving for x in the second equation, we get:

x=−6x = -6

Therefore, the x-intercepts of the graph of the function f(x)=(x+3)2−9f(x) = (x+3)^2 - 9 are x=0x = 0 and x=−6x = -6.

Conclusion

In this problem, we found the x-intercepts of the graph of the quadratic function f(x)=(x+3)2−9f(x) = (x+3)^2 - 9. We used the steps of setting the function equal to 0, simplifying the equation, and solving for x to find the x-intercepts. The x-intercepts are the points where the graph of the function crosses the x-axis, and they are the solutions to the equation f(x)=0f(x) = 0. In this case, the x-intercepts are x=0x = 0 and x=−6x = -6.

Answer

The correct answer is:

A. { (0,0) $}$ and { (-6,0) $}$

This answer is based on the x-intercepts we found in the previous section. The x-intercepts are the points where the graph of the function crosses the x-axis, and they are the solutions to the equation f(x)=0f(x) = 0. In this case, the x-intercepts are x=0x = 0 and x=−6x = -6.

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Understanding Quadratic Functions

A quadratic function is a polynomial function of degree two, which means the highest power of the variable (usually x) is two. The general form of a quadratic function is:

f(x)=ax2+bx+cf(x) = ax^2 + bx + c

where a, b, and c are constants, and a cannot be zero.

What are x-intercepts?

The x-intercepts of a quadratic function are the points where the graph of the function crosses the x-axis. In other words, the x-intercepts are the solutions to the equation f(x)=0f(x) = 0. These points are also known as the roots or zeros of the function.

Why are x-intercepts important?

X-intercepts are important because they provide valuable information about the behavior of the function. For example, the x-intercepts can help us determine the number of solutions to the equation f(x)=0f(x) = 0, which can be useful in various applications such as physics, engineering, and economics.

How to find x-intercepts?

To find the x-intercepts of a quadratic function, we can use the following steps:

  1. Set the function equal to 0: We start by setting the function equal to 0, which gives us the equation f(x)=0f(x) = 0.
  2. Simplify the equation: We then simplify the equation by expanding the squared term and combining like terms.
  3. Solve for x: Finally, we solve for x by isolating the variable x on one side of the equation.

Frequently Asked Questions (FAQs)

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

A: The x-intercepts are the points where the graph of the function crosses the x-axis, while the y-intercepts are the points where the graph of the function crosses the y-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 use the steps outlined above: set the function equal to 0, simplify the equation, and solve for x.

Q: What if the quadratic function has no real x-intercepts?

A: If the quadratic function has no real x-intercepts, it means that the graph of the function does not cross the x-axis. In this case, the function has complex x-intercepts, which are not real numbers.

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 the graph of a quadratic function is a parabola, which can intersect the x-axis at most two times.

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

A: To determine the number of x-intercepts of a quadratic function, you can use the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is positive, the function has two real x-intercepts. If the discriminant is zero, the function has one real x-intercept. If the discriminant is negative, the function has no real x-intercepts.

Conclusion

In this article, we have discussed the concept of x-intercepts of quadratic functions, including how to find them and why they are important. We have also answered some frequently asked questions about x-intercepts, including the difference between x-intercepts and y-intercepts, how to find x-intercepts, and what to do if the quadratic function has no real x-intercepts.

Additional Resources

For more information on quadratic functions and x-intercepts, you can consult the following resources:

  • Khan Academy: Quadratic Functions
  • Mathway: Quadratic Functions
  • Wolfram Alpha: Quadratic Functions

By following the steps outlined in this article and using the resources provided, you can gain a deeper understanding of quadratic functions and x-intercepts.