Given The Function F ( X ) = 3 X 2 − 8 X + 2 F(x) = 3x^2 - 8x + 2 F ( X ) = 3 X 2 − 8 X + 2 , Find The Following:- Vertex: \qquad - Y Y Y -intercept: \qquad - X X X -intercepts: \qquad - Maximum/Minimum: \qquad

Introduction

Quadratic functions are a fundamental concept in mathematics, and understanding their key features is crucial for solving various problems in algebra, calculus, and other branches of mathematics. In this article, we will explore the key features of a quadratic function, specifically the vertex, y-intercept, x-intercepts, and maximum/minimum values. We will use the function $f(x) = 3x^2 - 8x + 2$ as a case study to illustrate these concepts.

Vertex

The vertex of a quadratic function is the point at which the function changes from decreasing to increasing or vice versa. It is also the maximum or minimum point of the function, depending on the direction of the parabola. To find the vertex of a quadratic function in the form $f(x) = ax^2 + bx + c$, we can use the formula:

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

In our case, $a = 3$ and $b = -8$, so we can plug these values into the formula to find the x-coordinate of the vertex:

$$x = -\frac{-8}{2(3)} = \frac{8}{6} = \frac{4}{3}$$

To find the y-coordinate of the vertex, we can plug the x-coordinate back into the original function:

$$f\left(\frac{4}{3}\right) = 3\left(\frac{4}{3}\right)^2 - 8\left(\frac{4}{3}\right) + 2$$

Simplifying this expression, we get:

$$f\left(\frac{4}{3}\right) = 3\left(\frac{16}{9}\right) - \frac{32}{3} + 2$$

$$f\left(\frac{4}{3}\right) = \frac{16}{3} - \frac{32}{3} + 2$$

$$f\left(\frac{4}{3}\right) = -\frac{16}{3} + 2$$

$$f\left(\frac{4}{3}\right) = -\frac{16}{3} + \frac{6}{3}$$

$$f\left(\frac{4}{3}\right) = -\frac{10}{3}$$

Therefore, the vertex of the function $f(x) = 3x^2 - 8x + 2$ is at the point $\left(\frac{4}{3}, -\frac{10}{3}\right)$.

y-Intercept

The y-intercept of a quadratic function is the point at which the function intersects the y-axis. It is the value of the function when $x = 0$. To find the y-intercept of a quadratic function in the form $f(x) = ax^2 + bx + c$, we can plug $x = 0$ into the function:

$$f(0) = a(0)^2 + b(0) + c$$

$$f(0) = c$$

In our case, $c = 2$, so the y-intercept of the function $f(x) = 3x^2 - 8x + 2$ is at the point $(0, 2)$.

x-Intercepts

The x-intercepts of a quadratic function are the points at which the function intersects the x-axis. They are the values of $x$ for which $f(x) = 0$. To find the x-intercepts of a quadratic function in the form $f(x) = ax^2 + bx + c$, we can set the function equal to zero and solve for $x$:

$$ax^2 + bx + c = 0$$

In our case, we have:

$$3x^2 - 8x + 2 = 0$$

We can solve this quadratic equation using the quadratic formula:

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

Plugging in the values $a = 3$, $b = -8$, and $c = 2$, we get:

$$x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(3)(2)}}{2(3)}$$

$$x = \frac{8 \pm \sqrt{64 - 24}}{6}$$

$$x = \frac{8 \pm \sqrt{40}}{6}$$

$$x = \frac{8 \pm 2\sqrt{10}}{6}$$

$$x = \frac{4 \pm \sqrt{10}}{3}$$

Therefore, the x-intercepts of the function $f(x) = 3x^2 - 8x + 2$ are at the points $\left(\frac{4 + \sqrt{10}}{3}, 0\right)$ and $\left(\frac{4 - \sqrt{10}}{3}, 0\right)$.

Maximum/Minimum

The maximum or minimum value of a quadratic function is the value of the function at the vertex. Since the vertex is at the point $\left(\frac{4}{3}, -\frac{10}{3}\right)$, the maximum or minimum value of the function $f(x) = 3x^2 - 8x + 2$ is $-\frac{10}{3}$.

Conclusion

In this article, we have explored the key features of a quadratic function, specifically the vertex, y-intercept, x-intercepts, and maximum/minimum values. We have used the function $f(x) = 3x^2 - 8x + 2$ as a case study to illustrate these concepts. By understanding these key features, we can better analyze and solve problems involving quadratic functions.

Key Takeaways

• The vertex of a quadratic function is the point at which the function changes from decreasing to increasing or vice versa.
• The y-intercept of a quadratic function is the point at which the function intersects the y-axis.
• The x-intercepts of a quadratic function are the points at which the function intersects the x-axis.
• The maximum or minimum value of a quadratic function is the value of the function at the vertex.

Further Reading

For further reading on quadratic functions, we recommend the following resources:

• Khan Academy: Quadratic Functions
• Math Is Fun: Quadratic Functions
• Wolfram MathWorld: Quadratic Function

References

• [1] Khan Academy. Quadratic Functions. Retrieved from https://www.khanacademy.org/math/algebra/quadratic-equations
• [2] Math Is Fun. Quadratic Functions. Retrieved from https://www.mathisfun.com/algebra/quadratic-functions.html
• [3] Wolfram MathWorld. Quadratic Function. Retrieved from https://mathworld.wolfram.com/QuadraticFunction.html
  Quadratic Function Q&A ==========================

Introduction

Quadratic functions are a fundamental concept in mathematics, and understanding their key features is crucial for solving various problems in algebra, calculus, and other branches of mathematics. In this article, we will answer some frequently asked questions about quadratic functions, covering topics such as the vertex, y-intercept, x-intercepts, and maximum/minimum values.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point at which the function changes from decreasing to increasing or vice versa. It is also the maximum or minimum point of the function, depending on the direction of the parabola.

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

A: To find the vertex of a quadratic function in the form $f(x) = ax^2 + bx + c$, you can use the formula:

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

Then, plug the x-coordinate back into the original function to find the y-coordinate of the vertex.

Q: What is the y-intercept of a quadratic function?

A: The y-intercept of a quadratic function is the point at which the function intersects the y-axis. It is the value of the function when $x = 0$.

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

A: To find the y-intercept of a quadratic function in the form $f(x) = ax^2 + bx + c$, simply plug $x = 0$ into the function. The result will be the y-intercept.

Q: What are the x-intercepts of a quadratic function?

A: The x-intercepts of a quadratic function are the points at which the function intersects the x-axis. They are the values of $x$ for which $f(x) = 0$.

Q: How do I find the x-intercepts of a quadratic function?

A: To find the x-intercepts of a quadratic function in the form $f(x) = ax^2 + bx + c$, set the function equal to zero and solve for $x$ using the quadratic formula:

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

Q: What is the maximum or minimum value of a quadratic function?

A: The maximum or minimum value of a quadratic function is the value of the function at the vertex. Since the vertex is at the point $\left(\frac{4}{3}, -\frac{10}{3}\right)$, the maximum or minimum value of the function $f(x) = 3x^2 - 8x + 2$ is $-\frac{10}{3}$.

Q: How do I determine whether a quadratic function is increasing or decreasing?

A: To determine whether a quadratic function is increasing or decreasing, look at the coefficient of the $x^2$ term. If the coefficient is positive, the function is increasing. If the coefficient is negative, the function is decreasing.

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

A: Yes, a quadratic function can have more than one x-intercept. This occurs when the quadratic equation has two distinct real roots.

Q: Can a quadratic function have no x-intercepts?

A: Yes, a quadratic function can have no x-intercepts. This occurs when the quadratic equation has no real roots.

Conclusion

In this article, we have answered some frequently asked questions about quadratic functions, covering topics such as the vertex, y-intercept, x-intercepts, and maximum/minimum values. By understanding these key features, we can better analyze and solve problems involving quadratic functions.

Key Takeaways

• The vertex of a quadratic function is the point at which the function changes from decreasing to increasing or vice versa.
• The y-intercept of a quadratic function is the point at which the function intersects the y-axis.
• The x-intercepts of a quadratic function are the points at which the function intersects the x-axis.
• The maximum or minimum value of a quadratic function is the value of the function at the vertex.

Further Reading

For further reading on quadratic functions, we recommend the following resources:

• Khan Academy: Quadratic Functions
• Math Is Fun: Quadratic Functions
• Wolfram MathWorld: Quadratic Function

References

• [1] Khan Academy. Quadratic Functions. Retrieved from https://www.khanacademy.org/math/algebra/quadratic-equations
• [2] Math Is Fun. Quadratic Functions. Retrieved from https://www.mathisfun.com/algebra/quadratic-functions.html
• [3] Wolfram MathWorld. Quadratic Function. Retrieved from https://mathworld.wolfram.com/QuadraticFunction.html