Which Point Is An $x$-intercept Of The Quadratic Function $f(x) = (x+6)(x-3)$?A. $ ( 0 , − 6 ) (0, -6) ( 0 , − 6 ) [/tex] B. $(-6, 0)$ C. $(6, 0)$ D. $ ( 0 , 6 ) (0, 6) ( 0 , 6 ) [/tex]

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Quadratic functions are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we will delve into the concept of quadratic functions, specifically the function $f(x) = (x+6)(x-3)$, and explore the concept of x-intercepts.

What are Quadratic Functions?

A quadratic function is a polynomial function of degree two, which means the highest power of the variable (in this case, x) is two. The general form of a quadratic function is $f(x) = ax^2 + bx + c$, where a, b, and c are constants. Quadratic functions can be represented graphically as a parabola, which is a U-shaped curve.

The Given Quadratic Function

The given quadratic function is $f(x) = (x+6)(x-3)$. To understand this function, we can expand it using the distributive property:

f(x)=(x+6)(x−3)f(x) = (x+6)(x-3)

f(x)=x2−3x+6x−18f(x) = x^2 - 3x + 6x - 18

f(x)=x2+3x−18f(x) = x^2 + 3x - 18

Finding x-Intercepts

An x-intercept is a point on the graph of a function where the function intersects the x-axis. In other words, it is a point where the value of y is zero. To find the x-intercepts of a quadratic function, we need to set the function equal to zero and solve for x.

Let's set the given function equal to zero:

x2+3x−18=0x^2 + 3x - 18 = 0

We can solve this quadratic equation using various methods, such as factoring, the quadratic formula, or graphing. In this case, we can factor the equation:

(x+6)(x−3)=0(x + 6)(x - 3) = 0

This tells us that either $(x + 6) = 0$ or $(x - 3) = 0$.

Solving for x, we get:

x+6=0⇒x=−6x + 6 = 0 \Rightarrow x = -6

x−3=0⇒x=3x - 3 = 0 \Rightarrow x = 3

Therefore, the x-intercepts of the quadratic function $f(x) = (x+6)(x-3)$ are $( -6, 0)$ and $(3, 0)$.

Comparing with the Options

Now that we have found the x-intercepts of the quadratic function, let's compare them with the given options:

A. $(0, -6)$ B. $(-6, 0)$ C. $(6, 0)$ D. $(0, 6)$

From our previous calculation, we know that the x-intercepts are $( -6, 0)$ and $(3, 0)$. Therefore, the correct answer is:

B. $(-6, 0)$

Conclusion

In this article, we explored the concept of quadratic functions and x-intercepts. We analyzed the given quadratic function $f(x) = (x+6)(x-3)$ and found its x-intercepts using factoring. We then compared the x-intercepts with the given options and concluded that the correct answer is B. $(-6, 0)$.

Final Thoughts

Understanding quadratic functions and x-intercepts is crucial for solving various mathematical problems. By analyzing the given quadratic function and finding its x-intercepts, we can gain a deeper understanding of the properties of quadratic functions. This knowledge can be applied to various fields, such as physics, engineering, and economics, where quadratic functions are commonly used to model real-world phenomena.

References

  • [1] "Quadratic Functions" by Khan Academy
  • [2] "x-Intercepts" by Math Open Reference
  • [3] "Quadratic Equations" by Purplemath

Additional Resources

  • [1] "Quadratic Functions" by Wolfram MathWorld
  • [2] "x-Intercepts" by IXL
  • [3] "Quadratic Equations" by Mathway
    Quadratic Functions and x-Intercepts: A Q&A Guide =====================================================

In our previous article, we explored the concept of quadratic functions and x-intercepts. We analyzed the given quadratic function $f(x) = (x+6)(x-3)$ and found its x-intercepts using factoring. In this article, we will provide a Q&A guide to help you better understand quadratic functions and x-intercepts.

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which means the highest power of the variable (in this case, x) is two. The general form of a quadratic function is $f(x) = ax^2 + bx + c$, where a, b, and c are constants.

Q: What is an x-intercept?

A: An x-intercept is a point on the graph of a function where the function intersects the x-axis. In other words, it is a point where the value of y is zero.

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

A: To find the x-intercepts of a quadratic function, you need to set the function equal to zero and solve for x. You can use various methods, such as factoring, the quadratic formula, or graphing.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve quadratic equations. It is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

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

A: To use the quadratic formula, you need to plug in the values of a, b, and c from the quadratic function into the formula. Then, simplify the expression and solve for x.

Q: What are some common mistakes to avoid when finding x-intercepts?

A: Some common mistakes to avoid when finding x-intercepts include:

  • Not setting the function equal to zero
  • Not solving for x correctly
  • Not considering all possible solutions

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you can use various methods, such as plotting points, using a graphing calculator, or using a graphing software.

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

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

  • Modeling the trajectory of a projectile
  • Finding the maximum or minimum value of a function
  • Solving optimization problems
  • Modeling population growth or decline

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 given by:

b2−4acb^2 - 4ac

If the discriminant is positive, the quadratic function has two x-intercepts. If the discriminant is zero, the quadratic function has one x-intercept. If the discriminant is negative, the quadratic function has no x-intercepts.

Conclusion

In this article, we provided a Q&A guide to help you better understand quadratic functions and x-intercepts. We covered topics such as the definition of a quadratic function, the concept of x-intercepts, and how to find x-intercepts using factoring and the quadratic formula. We also discussed common mistakes to avoid and real-world applications of quadratic functions and x-intercepts.

Final Thoughts

Understanding quadratic functions and x-intercepts is crucial for solving various mathematical problems. By mastering these concepts, you can gain a deeper understanding of the properties of quadratic functions and apply them to real-world problems.

References

  • [1] "Quadratic Functions" by Khan Academy
  • [2] "x-Intercepts" by Math Open Reference
  • [3] "Quadratic Equations" by Purplemath

Additional Resources

  • [1] "Quadratic Functions" by Wolfram MathWorld
  • [2] "x-Intercepts" by IXL
  • [3] "Quadratic Equations" by Mathway