Consider The Given Function K ( X ) = 3 X 2 − 6 X − 1 K(x) = 3x^2 - 6x - 1 K ( X ) = 3 X 2 − 6 X − 1 .(a) Write The Function In Vertex Form.(b) Identify The Vertex.(c) Determine The X X X -intercept(s).(d) Determine The Y Y Y -intercept(s).(e) Sketch The Function.(f) Determine The

Introduction

In this article, we will explore the properties of a given quadratic function, $k(x) = 3x^2 - 6x - 1$. We will first rewrite the function in vertex form, which will allow us to identify the vertex of the parabola. Then, we will determine the $x$-intercept(s) and $y$-intercept(s) of the function. Finally, we will sketch the function and use the information obtained to answer additional questions.

(a) Write the function in vertex form

To write the function in vertex form, we need to complete the square. The vertex form of a quadratic function is given by:

$k(x) = a(x - h)^2 + k$

where $(h, k)$ is the vertex of the parabola.

First, we factor out the leading coefficient, $a = 3$:

$k(x) = 3(x^2 - 2x) - 1$

Next, we add and subtract the square of half the coefficient of the $x$ term, which is $-2$. We get:

$k(x) = 3(x^2 - 2x + 1 - 1) - 1$

Now, we can rewrite the expression as:

$k(x) = 3((x - 1)^2 - 1) - 1$

Expanding the expression, we get:

$k(x) = 3(x - 1)^2 - 3 - 1$

Simplifying, we get:

$k(x) = 3(x - 1)^2 - 4$

Therefore, the function in vertex form is:

$k(x) = 3(x - 1)^2 - 4$

(b) Identify the vertex

The vertex of the parabola is given by the point $(h, k)$. In this case, we can see that $h = 1$ and $k = -4$. Therefore, the vertex of the parabola is $(1, -4)$.

(c) Determine the $x$-intercept(s)

To determine the $x$-intercept(s), we need to set $y = 0$ and solve for $x$. We get:

$0 = 3x^2 - 6x - 1$

We can factor the quadratic expression as:

$0 = (3x + 1)(x - 1)$

Solving for $x$, we get:

$x = -\frac{1}{3}$ or $x = 1$

Therefore, the $x$-intercept(s) are $x = -\frac{1}{3}$ and $x = 1$.

(d) Determine the $y$-intercept(s)

To determine the $y$-intercept(s), we need to set $x = 0$ and solve for $y$. We get:

$y = 3(0)^2 - 6(0) - 1$

Simplifying, we get:

$y = -1$

Therefore, the $y$-intercept is $y = -1$.

(e) Sketch the function

To sketch the function, we can use the information obtained above. We know that the vertex of the parabola is $(1, -4)$, the $x$-intercept(s) are $x = -\frac{1}{3}$ and $x = 1$, and the $y$-intercept is $y = -1$.

Using this information, we can sketch the function as follows:

• The parabola opens upward, since the leading coefficient is positive.
• The vertex of the parabola is $(1, -4)$.
• The $x$-intercept(s) are $x = -\frac{1}{3}$ and $x = 1$.
• The $y$-intercept is $y = -1$.

(f) Determine the axis of symmetry

The axis of symmetry is the vertical line that passes through the vertex of the parabola. In this case, the axis of symmetry is the line $x = 1$.

Conclusion

In this article, we have explored the properties of the quadratic function $k(x) = 3x^2 - 6x - 1$. We have rewritten the function in vertex form, identified the vertex, determined the $x$-intercept(s) and $y$-intercept(s), sketched the function, and determined the axis of symmetry. We have used the information obtained to answer additional questions and provide a clear understanding of the properties of the function.

References

• [1] "Quadratic Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/quadratic.html
• [2] "Vertex Form of a Quadratic Function" by Purplemath. Retrieved from https://www.purplemath.com/modules/quadform.htm

Keywords

• Quadratic function
• Vertex form
• Vertex
• $x$-intercept(s)
• $y$-intercept(s)
• Axis of symmetry
• Quadratic equation
• Completing the square
  Quadratic Function Q&A =========================

Introduction

In this article, we will answer some frequently asked questions about quadratic functions. We will cover topics such as vertex form, vertex, $x$-intercept(s), $y$-intercept(s), and axis of symmetry.

Q1: What is the vertex form of a quadratic function?

A1: The vertex form of a quadratic function is given by:

$k(x) = a(x - h)^2 + k$

where $(h, k)$ is the vertex of the parabola.

Q2: How do I find the vertex of a quadratic function?

A2: To find the vertex of a quadratic function, you can use the formula:

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

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

Q3: What is the axis of symmetry of a quadratic function?

A3: 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:

$x = h$

where $h$ is the $x$-coordinate of the vertex.

Q4: How do I find the $x$-intercept(s) of a quadratic function?

A4: To find the $x$-intercept(s) of a quadratic function, you can set $y = 0$ and solve for $x$. This will give you the $x$-coordinate(s) of the $x$-intercept(s).

Q5: How do I find the $y$-intercept of a quadratic function?

A5: To find the $y$-intercept of a quadratic function, you can set $x = 0$ and solve for $y$. This will give you the $y$-coordinate of the $y$-intercept.

Q6: What is the difference between a quadratic function and a linear function?

A6: A quadratic function is a polynomial function of degree 2, while a linear function is a polynomial function of degree 1. Quadratic functions have a parabolic shape, while linear functions have a straight line shape.

Q7: How do I determine if a quadratic function is positive or negative?

A7: To determine if a quadratic function is positive or negative, you can look at the sign of the leading coefficient. If the leading coefficient is positive, the function is positive. If the leading coefficient is negative, the function is negative.

Q8: Can a quadratic function have more than one $x$-intercept?

A8: Yes, a quadratic function can have more than one $x$-intercept. This occurs when the quadratic function has a double root.

Q9: Can a quadratic function have more than one $y$-intercept?

A9: No, a quadratic function can only have one $y$-intercept.

Q10: How do I graph a quadratic function?

A10: To graph a quadratic function, you can use the following steps:

1. Find the vertex of the parabola.
2. Find the $x$-intercept(s) of the parabola.
3. Find the $y$-intercept of the parabola.
4. Plot the vertex, $x$-intercept(s), and $y$-intercept on a coordinate plane.
5. Draw a smooth curve through the points to form the parabola.

Conclusion

In this article, we have answered some frequently asked questions about quadratic functions. We have covered topics such as vertex form, vertex, $x$-intercept(s), $y$-intercept(s), and axis of symmetry. We hope that this article has been helpful in providing a clear understanding of quadratic functions.

References

• [1] "Quadratic Functions" by Math Open Reference. Retrieved from https://www.mathopenref.com/quadratic.html
• [2] "Vertex Form of a Quadratic Function" by Purplemath. Retrieved from https://www.purplemath.com/modules/quadform.htm

Keywords

• Quadratic function
• Vertex form
• Vertex
• $x$-intercept(s)
• $y$-intercept(s)
• Axis of symmetry
• Quadratic equation
• Completing the square
• Graphing quadratic functions