Consider The Given Function: $\[ F(x) = X^2 - 14x - 72 \\]What Are The Zeros And The Axis Of Symmetry For The Graph Of The Function?A. Zeros: \[$x = -4\$\] And \[$x = 18\$\]; Axis Of Symmetry: \[$x = 7\$\] B. Zeros:

Introduction

In mathematics, quadratic equations are a fundamental concept that plays a crucial role in various fields, including algebra, geometry, and calculus. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. In this article, we will focus on solving quadratic equations of the form $f(x) = ax^2 + bx + c$ and finding the zeros and axis of symmetry for the graph of the function.

The Given Function

The given function is $f(x) = x^2 - 14x - 72$. To find the zeros and axis of symmetry, we need to factorize the quadratic expression.

Factoring the Quadratic Expression

To factorize the quadratic expression, we need to find two numbers whose product is $-72$ and whose sum is $-14$. These numbers are $-18$ and $4$, because $-18 \times 4 = -72$ and $-18 + 4 = -14$. Therefore, we can write the quadratic expression as:

$f(x) = (x - 18)(x + 4)$

Finding the Zeros

The zeros of a quadratic function are the values of $x$ that make the function equal to zero. In other words, we need to find the values of $x$ that satisfy the equation $f(x) = 0$. To do this, we can set each factor equal to zero and solve for $x$.

$(x - 18) = 0 \Rightarrow x = 18$

$(x + 4) = 0 \Rightarrow x = -4$

Therefore, the zeros of the function are $x = -4$ and $x = 18$.

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 of a parabola is the point where the parabola changes direction. To find the axis of symmetry, we need to find the value of $x$ that makes the function equal to its average value. This value is given by the formula:

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

In this case, $a = 1$ and $b = -14$. Therefore, the axis of symmetry is:

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

Therefore, the axis of symmetry is $x = 7$.

Conclusion

In conclusion, we have found the zeros and axis of symmetry for the graph of the function $f(x) = x^2 - 14x - 72$. The zeros are $x = -4$ and $x = 18$, and the axis of symmetry is $x = 7$. These values are essential in understanding the behavior of the function and its graph.

Answer

The correct answer is:

A. Zeros: $x = -4$ and $x = 18$; Axis of symmetry: $x = 7$

Discussion

The given function is a quadratic function of the form $f(x) = ax^2 + bx + c$. To find the zeros and axis of symmetry, we need to factorize the quadratic expression and solve for $x$. The zeros of a quadratic function are the values of $x$ that make the function equal to zero, and the axis of symmetry is a vertical line that passes through the vertex of the parabola.

Example

Consider the function $f(x) = x^2 + 6x + 8$. To find the zeros and axis of symmetry, we need to factorize the quadratic expression and solve for $x$.

Step 1: Factorize the Quadratic Expression

To factorize the quadratic expression, we need to find two numbers whose product is $8$ and whose sum is $6$. These numbers are $4$ and $2$, because $4 \times 2 = 8$ and $4 + 2 = 6$. Therefore, we can write the quadratic expression as:

$f(x) = (x + 4)(x + 2)$

Step 2: Find the Zeros

The zeros of a quadratic function are the values of $x$ that make the function equal to zero. In other words, we need to find the values of $x$ that satisfy the equation $f(x) = 0$. To do this, we can set each factor equal to zero and solve for $x$.

$(x + 4) = 0 \Rightarrow x = -4$

$(x + 2) = 0 \Rightarrow x = -2$

Therefore, the zeros of the function are $x = -4$ and $x = -2$.

Step 3: Find 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 of a parabola is the point where the parabola changes direction. To find the axis of symmetry, we need to find the value of $x$ that makes the function equal to its average value. This value is given by the formula:

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

In this case, $a = 1$ and $b = 6$. Therefore, the axis of symmetry is:

$x = \frac{-6}{2(1)} = \frac{-6}{2} = -3$

Therefore, the axis of symmetry is $x = -3$.

Conclusion

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Introduction

Quadratic equations are a fundamental concept in mathematics that plays a crucial role in various fields, including algebra, geometry, and calculus. In this article, we will provide a comprehensive Q&A guide to help you understand quadratic equations and their applications.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the following methods:

1. Factoring: If the quadratic expression can be factored into the product of two binomials, you can set each factor equal to zero and solve for $x$.
2. Quadratic Formula: If the quadratic expression cannot be factored, you can use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
3. Graphing: You can also graph the quadratic function and find the x-intercepts, which represent the zeros of the function.

Q: What is the axis of symmetry?

A: The axis of symmetry is a vertical line that passes through the vertex of the parabola. The vertex of a parabola is the point where the parabola changes direction. To find the axis of symmetry, you can use the formula: $x = \frac{-b}{2a}$.

Q: How do I find the vertex of a parabola?

A: To find the vertex of a parabola, you can use the following methods:

1. Vertex Formula: If the quadratic expression is in the form $ax^2 + bx + c$, you can use the vertex formula: $x = \frac{-b}{2a}$.
2. Graphing: You can also graph the quadratic function and find the vertex, which is the point where the parabola changes direction.

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

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. In other words, a quadratic equation has a squared variable, while a linear equation does not.

Q: Can I use a calculator to solve a quadratic equation?

A: Yes, you can use a calculator to solve a quadratic equation. Most calculators have a built-in quadratic formula function that you can use to solve the equation.

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

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

1. Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
2. Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
3. Computer Science: Quadratic equations are used in algorithms and data structures, such as sorting and searching.
4. Economics: Quadratic equations are used to model economic systems and make predictions about future trends.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics that plays a crucial role in various fields. By understanding quadratic equations and their applications, you can solve problems and make predictions in a wide range of fields. We hope this Q&A guide has been helpful in understanding quadratic equations and their applications.