Select The Correct Answer From Each Drop-down Menu.The Graph Of The Function Y = X 2 ( X 2 − 6 X + 9 Y=x^2(x^2-6x+9 Y = X 2 ( X 2 − 6 X + 9 ] Has Zeros Of □ \square □ . So The Function Has □ \square □ Distinct Real Zeros And □ \square □ Complex Zeros.

Mar 9, 2025 by ADMIN 252 views

**Understanding the Graph of a Quadratic Function**

Introduction

In mathematics, 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 $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. The graph of a quadratic function is a parabola, which is a U-shaped curve. In this article, we will discuss the graph of the function $y=x^2(x^2-6x+9)$ and find its zeros.

The Graph of the Function

The given function is $y=x^2(x^2-6x+9)$ . To find the zeros of this function, we need to find the values of $x$ that make the function equal to zero. We can start by factoring the function:

y=x^2(x^2-6x+9)

y=x^2(x-3)^2

Now, we can see that the function has two factors: $x^2$ and $(x-3)^2$ . The first factor, $x^2$ , is equal to zero when $x=0$ . The second factor, $(x-3)^2$ , is equal to zero when $x=3$ .

Zeros of the Function

The zeros of the function are the values of $x$ that make the function equal to zero. In this case, the zeros of the function are $x=0$ and $x=3$ . These are the values of $x$ that make the function equal to zero.

Distinct Real Zeros

A distinct real zero is a zero that is a real number and is not repeated. In this case, the function has two distinct real zeros: $x=0$ and $x=3$ .

Complex Zeros

A complex zero is a zero that is a complex number. In this case, the function has no complex zeros, because the zeros of the function are real numbers.

Conclusion

In conclusion, the graph of the function $y=x^2(x^2-6x+9)$ has zeros of $x=0$ and $x=3$ . The function has two distinct real zeros and no complex zeros.

Select the Correct Answer from Each Drop-Down Menu

What are the zeros of the function $y=x^2(x^2-6x+9)$ $y = x^{2} (x^{2} - 6 x + 9)$ ?
- $x=0$ and $x=3$
- $x=0$ and $x=4$
- $x=1$ and $x=2$
- $x=2$ and $x=5$
How many distinct real zeros does the function have?
- 1
- 2
- 3
- 4
How many complex zeros does the function have?
- 1
- 2
- 3
- 0

Answer Key

What are the zeros of the function $y=x^2(x^2-6x+9)$ $y = x^{2} (x^{2} - 6 x + 9)$ ?
- $x=0$ and $x=3$
How many distinct real zeros does the function have?
- 2
How many complex zeros does the function have?
- 0
  Frequently Asked Questions (FAQs) about the Graph of a Quadratic Function ====================================================================

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 is two. The general form of a quadratic function is $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants.

Q: What is the graph of a quadratic function?

A: The graph of a quadratic function is a parabola, which is a U-shaped curve. The parabola can open upwards or downwards, depending on the value of $a$ in the quadratic function.

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

A: To find the zeros of a quadratic function, you need to find the values of $x$ that make the function equal to zero. You can do this by factoring the function, if possible, or by using the quadratic formula.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to find the zeros of a quadratic function. The formula is:

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

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of $a$ , $b$ , and $c$ from the quadratic function into the formula. Then, you can simplify the expression and find the values of $x$ that make the function equal to zero.

Q: What are the zeros of the function $y=x^2(x^2-6x+9)$ ?

A: The zeros of the function $y=x^2(x^2-6x+9)$ are $x=0$ and $x=3$ .

Q: How many distinct real zeros does the function $y=x^2(x^2-6x+9)$ have?

A: The function $y=x^2(x^2-6x+9)$ has two distinct real zeros.

Q: How many complex zeros does the function $y=x^2(x^2-6x+9)$ have?

A: The function $y=x^2(x^2-6x+9)$ has no complex zeros.

Q: What is the difference between a distinct real zero and a complex zero?

A: A distinct real zero is a zero that is a real number and is not repeated. A complex zero is a zero that is a complex number.

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

A: No, a quadratic function can have at most two zeros.

Q: Can a quadratic function have no zeros?

A: Yes, a quadratic function can have no zeros if the function is always positive or always negative.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you can use a graphing calculator or a computer program. You can also use a table of values to find the zeros of the function and then plot the points on a coordinate plane.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point on the graph where the function changes from decreasing to increasing or vice versa. The vertex can be found using the formula:

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

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

A: To find the vertex of a quadratic function, you need to plug in the values of $a$ and $b$ from the quadratic function into the formula. Then, you can simplify the expression and find the value of $x$ that corresponds to the vertex.

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 function. The equation of the axis of symmetry is:

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 need to plug in the values of $a$ and $b$ from the quadratic function into the formula. Then, you can simplify the expression and find the value of $x$ that corresponds to the axis of symmetry.