Select The Correct Answer.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.

Understanding the Problem

To find the zeros and the 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

To find the zeros of the function, we need to solve the equation $f(x) = 0$. This means 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:

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

Now, we can set each factor equal to zero and solve for $x$:

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

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

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

Using the Quadratic Formula

Alternatively, we can use the quadratic formula to find the zeros of the function. The quadratic formula is given by:

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

In this case, $a = 1$, $b = -14$, and $c = -72$. Plugging these values into the quadratic formula, we get:

$$x = \frac{-(-14) \pm \sqrt{(-14)^2 - 4(1)(-72)}}{2(1)}$$

Simplifying, we get:

$$x = \frac{14 \pm \sqrt{196 + 288}}{2}$$

$$x = \frac{14 \pm \sqrt{484}}{2}$$

$$x = \frac{14 \pm 22}{2}$$

So, the zeros of the function are $x = \frac{14 + 22}{2} = 18$ and $x = \frac{14 - 22}{2} = -4$.

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. The x-coordinate of the vertex is given by:

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

In this case, $a = 1$ and $b = -14$. Plugging these values into the formula, we get:

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

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

$$x = 7$$

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

Conclusion

In conclusion, the zeros of the function $f(x) = x^2 - 14x - 72$ are $x = -4$ and $x = 18$, and the axis of symmetry is $x = 7$. These values can be used to graph the function and find the vertex of the parabola.

Answer

The correct answer is:

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

Understanding Quadratic Functions

Quadratic functions are a type of polynomial function that can be written in the form $f(x) = ax^2 + bx + c$. These functions have a parabolic shape and can be used to model a wide range of real-world phenomena.

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: There are several ways to find the zeros of a quadratic function, including factoring, using the quadratic formula, and graphing. Factoring involves expressing the quadratic expression as a product of two binomials, while the quadratic formula involves using the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the solutions.

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 given by the equation $x = \frac{-b}{2a}$.

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 $x = \frac{-b}{2a}$. This will give you the x-coordinate of the vertex of the parabola.

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

A: The zeros of a quadratic function are symmetric about the axis of symmetry. In other words, if one zero is to the left of the axis of symmetry, the other zero will be to the right, and vice versa.

Q: Can you give an example of a quadratic function and its zeros and axis of symmetry?

A: Let's consider the quadratic function $f(x) = x^2 - 14x - 72$. To find the zeros, we can factor the quadratic expression as $(x - 18)(x + 4)$. Setting each factor equal to zero, we get $x = 18$ and $x = -4$. The axis of symmetry is given by $x = \frac{-b}{2a} = \frac{-(-14)}{2(1)} = 7$.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you can use the zeros and the axis of symmetry to draw the parabola. Start by plotting the zeros on the x-axis, then draw a vertical line through the axis of symmetry. The parabola will be symmetric about this line.

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

A: Quadratic functions have many real-world applications, including modeling the trajectory of a projectile, the motion of an object under the influence of gravity, and the growth of a population.

Q: Can you give some examples of quadratic functions in real-world applications?

A: Here are a few examples:

• The trajectory of a baseball thrown from a height of 3 meters with an initial velocity of 20 meters per second can be modeled using the quadratic function $f(x) = -5x^2 + 20x + 3$.
• The motion of a car traveling down a hill can be modeled using the quadratic function $f(x) = -2x^2 + 10x + 5$.
• The growth of a population can be modeled using the quadratic function $f(x) = 2x^2 + 5x + 1$.

Conclusion

In conclusion, quadratic functions are a powerful tool for modeling real-world phenomena. By understanding the zeros and axis of symmetry of a quadratic function, you can graph the function and use it to make predictions about the behavior of the system being modeled.