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

Understanding the Problem

To find the zeros and axis of symmetry for the graph of a quadratic function, we need to use the given function $f(x) = x^2 - 14x - 72$. The zeros of a function are the values of $x$ where the function intersects the x-axis, and the axis of symmetry is the vertical line that passes through the vertex of the parabola.

Finding the Zeros of the Function

To find the zeros of the function, we need to solve the equation $f(x) = 0$. This means that we need to find the values of $x$ that make the function equal to zero. We can do this by factoring the quadratic expression or by using the quadratic formula.

Factoring the Quadratic Expression

Let's try to factor the quadratic expression $x^2 - 14x - 72$. We can start by finding 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 $(x - 18)(x + 4)$.

Solving for the Zeros

Now that we have factored the quadratic expression, we can set each factor equal to zero and solve for $x$. This gives us the following equations:

$x - 18 = 0$ and $x + 4 = 0$

Solving for $x$ in each equation, we get:

$x = 18$ and $x = -4$

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

Finding the Axis of Symmetry

The axis of symmetry is the vertical line that passes through the vertex of the parabola. To find the axis of symmetry, we need to find the x-coordinate of the vertex. We can do this by using the formula $x = \frac{-b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic expression.

Finding the x-Coordinate of the Vertex

In the given function $f(x) = x^2 - 14x - 72$, the coefficient of $x^2$ is $1$ and the coefficient of $x$ is $-14$. Therefore, we can plug these values into the formula to get:

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

Simplifying the expression, we get:

$x = \frac{14}{2}$

$x = 7$

Therefore, the x-coordinate of the vertex is $x = 7$.

Finding the Axis of Symmetry

Since the x-coordinate of the vertex is $x = 7$, the axis of symmetry is the vertical line that passes through this point. Therefore, the axis of symmetry is $x = 7$.

Conclusion

In conclusion, the zeros of the function $f(x) = x^2 - 14x - 72$ are $x = 18$ and $x = -4$, and the axis of symmetry is $x = 7$.

Answer

The correct answer is:

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

However, this answer is incorrect. The correct zeros are $x = 18$ and $x = -4$, and the correct axis of symmetry is $x = 7$.

Understanding Quadratic Functions

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.

Q: What are the zeros of a quadratic function?

A: The zeros of a quadratic function are the values of $x$ where the function intersects the x-axis. In other words, they are the solutions to the equation $f(x) = 0$.

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

A: To find the zeros of a quadratic function, you can use factoring, the quadratic formula, or graphing. Factoring involves expressing the quadratic expression as a product of two binomials, while 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 the vertical line that passes through the vertex of the parabola. It is the line that divides the parabola into two equal parts.

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

A: To find the axis of symmetry of a quadratic function, you can use the formula $x = \frac{-b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic expression.

Q: What is the relationship between the zeros and axis of symmetry of a quadratic function?

A: The zeros of a quadratic function are symmetric about the axis of symmetry. This means that if one zero is to the left of the axis of symmetry, the other zero is to the right, and vice versa.

Q: Can a quadratic function have more than two zeros?

A: No, a quadratic function can have at most two zeros. This is because a quadratic function is a polynomial of degree two, and the Fundamental Theorem of Algebra states that a polynomial of degree $n$ has at most $n$ zeros.

Q: Can a quadratic function have no zeros?

A: Yes, a quadratic function can have no zeros. This occurs when the quadratic expression is always positive or always negative, and never intersects the x-axis.

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

A: To determine the number of zeros of a quadratic function, you can use the discriminant, which is the expression $b^2 - 4ac$. If the discriminant is positive, the quadratic function has two distinct zeros. If the discriminant is zero, the quadratic function has one zero. If the discriminant is negative, the quadratic function has no zeros.

Q: Can a quadratic function have a zero that is not an integer?

A: Yes, a quadratic function can have a zero that is not an integer. This occurs when the quadratic expression is not factorable into integers, and the zeros are irrational numbers.

Conclusion

In conclusion, the zeros and axis of symmetry of a quadratic function are important concepts in algebra. Understanding these concepts can help you solve quadratic equations and graph quadratic functions.

Frequently Asked Questions

• Q: What is the difference between a quadratic function and a linear function? A: A quadratic function is a polynomial of degree two, while a linear function is a polynomial of degree one.
• Q: Can a quadratic function be a perfect square? A: Yes, a quadratic function can be a perfect square. For example, the quadratic function $f(x) = x^2 + 4x + 4$ is a perfect square.
• Q: Can a quadratic function have a zero that is a complex number? A: Yes, a quadratic function can have a zero that is a complex number. This occurs when the discriminant is negative.

Additional Resources

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