Select The Correct Answer.Consider The Given Function:$ F(x) = X^2 - 14x - 72 }$What Are The Zeros And Axis Of Symmetry For The Graph Of The Function?A. Zeros { X = -9$ $ And { X = -8$} ; A X I S O F S Y M M E T R Y : \[ ; Axis Of Symmetry: \[ ; A X I So F Sy Mm E T Ry : \[ X =

Understanding Quadratic Functions

Quadratic functions are a fundamental concept in mathematics, and they play a crucial role in various fields, including physics, engineering, and economics. A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is:

${ f(x) = ax^2 + bx + c }$

where ${ a }$, ${ b }$, and ${ c }$ are constants, and ${ x }$ is the variable.

The Given Function

In this article, we will be working with the following quadratic function:

${ f(x) = x^2 - 14x - 72 }$

Our goal is to find the zeros and axis of symmetry for the graph of this function.

Finding Zeros

The zeros of a quadratic function are the values of ${ x }$ that make the function equal to zero. In other words, they are the solutions to the equation:

${ f(x) = 0 }$

To find the zeros of the given function, we can use the factoring method or the quadratic formula. Let's use the factoring method.

Factoring the Quadratic Expression

We can factor the quadratic expression as follows:

${ x^2 - 14x - 72 = (x - 18)(x + 4) }$

Finding the Zeros

Now that we have factored the quadratic expression, we can find the zeros by setting each factor equal to zero and solving for ${ x }$.

${ x - 18 = 0 \quad \Rightarrow \quad x = 18 }$

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

Therefore, the zeros of the given function are:

${ x = 18 \quad \text{and} \quad x = -4 }$

Finding the Axis of Symmetry

The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the parabola. The vertex is the point on the parabola that is equidistant from the zeros.

To find the axis of symmetry, we can use the formula:

${ x = \frac{-b}{2a} }$

where ${ a }$ and ${ b }$ are the coefficients of the quadratic function.

In this case, ${ a = 1 }$ and ${ b = -14 }$, so we have:

${ x = \frac{-(-14)}{2(1)} = \frac{14}{2} = 7 }$

Therefore, the axis of symmetry is:

${ x = 7 }$

Conclusion

In this article, we have found the zeros and axis of symmetry for the graph of the given quadratic function. The zeros are ${ x = 18 }$ and ${ x = -4 }$, and the axis of symmetry is ${ x = 7 }$. These values are essential in understanding the behavior of the function and its graph.

Real-World Applications

Quadratic functions have numerous real-world applications, including:

• Physics: Quadratic functions are used to model the motion of objects under the influence of gravity, friction, and other forces.
• Engineering: Quadratic functions are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
• Economics: Quadratic functions are used to model economic systems, including supply and demand curves, and cost-benefit analysis.

Final Thoughts

In conclusion, quadratic functions are a fundamental concept in mathematics, and they have numerous real-world applications. By understanding how to find the zeros and axis of symmetry of a quadratic function, we can gain valuable insights into the behavior of the function and its graph.

Understanding Quadratic Functions

Quadratic functions are a fundamental concept in mathematics, and they play a crucial role in various fields, including physics, engineering, and economics. A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is:

${ f(x) = ax^2 + bx + c }$

where ${ a }$, ${ b }$, and ${ c }$ are constants, and ${ x }$ is the variable.

Quadratic Functions: Q&A

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

A: A linear function is a polynomial function of degree one, which means the highest power of the variable is one. A quadratic function, on the other hand, is a polynomial function of degree two, which means the highest power of the variable is two.

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

A: To find the zeros of a quadratic function, you can use the factoring method or the quadratic formula. The factoring method involves factoring the quadratic expression into two binomials, and then setting each binomial equal to zero and solving for ${ x }$. The quadratic formula involves using the formula:

${ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} }$

Q: What is the axis of symmetry of a quadratic function?

A: The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the parabola. The vertex is the point on the parabola that is equidistant from the zeros. To find the axis of symmetry, you can use the formula:

${ x = \frac{-b}{2a} }$

Q: How do I determine the direction of the parabola?

A: To determine the direction of the parabola, you can look at the coefficient of the ${ x^2 }$ term. If the coefficient is positive, the parabola opens upward. If the coefficient is negative, the parabola opens downward.

Q: Can I use a calculator to find the zeros and axis of symmetry of a quadratic function?

A: Yes, you can use a calculator to find the zeros and axis of symmetry of a quadratic function. Most graphing calculators have a built-in function to find the zeros and axis of symmetry of a quadratic function.

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

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

• Physics: Quadratic functions are used to model the motion of objects under the influence of gravity, friction, and other forces.
• Engineering: Quadratic functions are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
• Economics: Quadratic functions are used to model economic systems, including supply and demand curves, and cost-benefit analysis.

Q: Can I use quadratic functions to model real-world data?

A: Yes, you can use quadratic functions to model real-world data. Quadratic functions can be used to model data that exhibits a parabolic shape, such as the growth of a population or the decay of a substance.

Conclusion

In conclusion, quadratic functions are a fundamental concept in mathematics, and they have numerous real-world applications. By understanding how to find the zeros and axis of symmetry of a quadratic function, you can gain valuable insights into the behavior of the function and its graph. Whether you are a student, a teacher, or a professional, quadratic functions are an essential tool to have in your mathematical toolkit.

Final Thoughts

Quadratic functions are a powerful tool for modeling and analyzing real-world data. By understanding how to use quadratic functions, you can gain a deeper understanding of the world around you and make more informed decisions. Whether you are interested in physics, engineering, economics, or any other field, quadratic functions are an essential part of your mathematical toolkit.