The Given Quadratic Function

Oct 13, 2025 by thesweetdesigns.com 25 views

Quadratic functions are a fundamental concept in mathematics, and understanding them is crucial for solving various problems in algebra, geometry, and other branches of mathematics. In this article, we will focus on quadratic functions of the form ${$ f(x) = (x-a)(x-b) $}$, where ${$ a $}$ and ${$ b $}$ are constants. We will also explore the concept of ${$ x $}$-intercepts and learn how to identify them in a quadratic function.

What are Quadratic Functions?

A quadratic function is a polynomial function of degree two, which means the highest power of the variable ${$ x $}$ is two. The general form of a quadratic function is ${$ f(x) = ax^2 + bx + c $}$, where ${$ a $}$, ${$ b $}$, and ${$ c $}$ are constants, and ${$ a $}$ cannot be zero. Quadratic functions can be represented graphically as parabolas, which are U-shaped curves.

The Given Quadratic Function

The given quadratic function is ${$ f(x) = (x-4)(x+2) $}$. To understand this function, we need to expand it and rewrite it in the standard form ${$ f(x) = ax^2 + bx + c $}$. Expanding the given function, we get:

${$ f(x) = (x-4)(x+2) $}{$ f(x) = x^2 + 2x - 4x - 8 $}{$ f(x) = x^2 - 2x - 8 $}$

Now, we can see that the given quadratic function is in the form ${$ f(x) = ax^2 + bx + c $}$, where ${$ a = 1 $}$, ${$ b = -2 $}$, and ${$ c = -8 $}$.

What are ${$ x $}$-Intercepts?

An ${$ x $}$-intercept is a point on the graph of a function where the function crosses the ${$ x $}$-axis. In other words, it is a point where the value of the function is zero. For a quadratic function, the ${$ x $}$-intercepts are the points where the parabola intersects the ${$ x $}$-axis.

Finding ${$ x $}$-Intercepts

To find the ${$ x $}$-intercepts of a quadratic function, we need to set the function equal to zero and solve for ${$ x $}$. In the case of the given quadratic function, we have:

${$ f(x) = x^2 - 2x - 8 = 0 $}$

We can solve this equation by factoring, completing the square, or using the quadratic formula. Let's use factoring to solve this equation:

${$ (x-4)(x+2) = 0 $}$

This equation is true when either ${$ x-4 = 0 $}$ or ${$ x+2 = 0 $}$. Solving for ${$ x $}$ in both cases, we get:

${$ x-4 = 0 \Rightarrow x = 4 $}{$ x+2 = 0 \Rightarrow x = -2 $}$

Therefore, the ${$ x $}$-intercepts of the given quadratic function are ${$ (4,0) $}$ and ${$ (-2,0) $}$.

Conclusion

In this article, we learned about quadratic functions and their ${$ x $}$-intercepts. We explored the concept of quadratic functions, expanded the given quadratic function, and found its ${$ x $}$-intercepts. We also discussed the importance of understanding quadratic functions and their applications in various branches of mathematics.

Answer

Based on the discussion above, the correct answer is:

${$ \boxed{B. (-2,0)} $}$

$This is because the given quadratic function has an ${$ x \$}$-intercept at ${$ (-2,0) \$}$.<br/> **Quadratic Functions and Intercepts: A Q&A Guide**$

In our previous article, we explored the concept of quadratic functions and their ${$ x $}$-intercepts. We learned how to identify and find the ${$ x $}$-intercepts of a quadratic function. In this article, we will answer some frequently asked questions about quadratic functions and their ${$ 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 ${$ x $}$ is two. The general form of a quadratic function is ${$ f(x) = ax^2 + bx + c $}$, where ${$ a $}$, ${$ b $}$, and ${$ c $}$ are constants, and ${$ a $}$ cannot be zero.

Q: How do I identify a quadratic function?

A: To identify a quadratic function, look for the following characteristics:

The function has a highest power of two, which means the variable ${$ x $}$ is squared.
The function has a constant term, which is the value of the function when ${$ x $}$ is zero.
The function can be written in the form ${$ f(x) = ax^2 + bx + c $}$, where ${$ a $}$, ${$ b $}$, and ${$ c $}$ are constants.

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

A: To find the ${$ x $}$-intercepts of a quadratic function, set the function equal to zero and solve for ${$ x $}$. You can use factoring, completing the square, or the quadratic formula to solve the equation.

Q: What is the difference between a quadratic function and a linear function?

A: A quadratic function is a polynomial function of degree two, while a linear function is a polynomial function of degree one. In other words, a quadratic function has a highest power of two, while a linear function has a highest power of one.

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 parabola, and a parabola can intersect the ${$ x $}$-axis at most two times.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, follow these steps:

Find the ${$ x $}$-intercepts of the function.
Find the vertex of the parabola.
Plot the vertex and the ${$ x $}$-intercepts on a coordinate plane.
Draw a smooth curve through the points to form the parabola.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the highest or lowest point on the parabola. It is the point where the parabola changes direction.

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

A: To find the vertex of a quadratic function, use the formula ${$ x = -\frac{b}{2a} $}$ to find the ${$ x $}$-coordinate of the vertex. Then, substitute this value into the function to find the ${$ y $}$-coordinate of the vertex.

Conclusion

In this article, we answered some frequently asked questions about quadratic functions and their ${$ x $}$-intercepts. We learned how to identify a quadratic function, find the ${$ x $}$-intercepts of a quadratic function, and graph a quadratic function. We also discussed the vertex of a quadratic function and how to find it.

Additional Resources

If you want to learn more about quadratic functions and their ${$ x $}$-intercepts, check out the following resources:

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

Practice Problems

Try these practice problems to test your understanding of quadratic functions and their ${$ x $}$-intercepts:

Find the ${$ x $}$-intercepts of the quadratic function ${$ f(x) = x^2 + 4x + 4 $}$.
Graph the quadratic function ${$ f(x) = x^2 - 6x + 8 $].
Find the vertex of the quadratic function [$ f(x) = x^2 - 2x - 3 $].

Answer Key

[$ (-2,0) $}$
The graph of the quadratic function is a parabola that opens upward.
The vertex of the quadratic function is ${$ (1,-2) $}$.