What Are The $x$-intercept And Vertex Of This Quadratic Function?$\[ G(x) = -5(x-3)^2 \\]Write Each Feature As An Ordered Pair: \[$(a, B)\$\].The $x$-intercept Of Function $g$ Is

What are the $x$-intercept and vertex of this quadratic function?

Understanding Quadratic Functions

Quadratic functions are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. 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, and $a$ cannot be zero.

The Given Quadratic Function

In this article, we are given a quadratic function $g(x) = -5(x-3)^2$. Our goal is to find the $x$-intercept and vertex of this function. To do this, we need to understand the properties of quadratic functions and how to find their key features.

The $x$-intercept of a Quadratic Function

The $x$-intercept of a quadratic function is the point where the function intersects the $x$-axis. In other words, it is the point where the function has a value of zero. To find the $x$-intercept of a quadratic function, we need to set the function equal to zero and solve for $x$.

Finding the $x$-intercept of $g(x)$

To find the $x$-intercept of $g(x) = -5(x-3)^2$, we need to set the function equal to zero and solve for $x$. This gives us the equation:

$$-5(x-3)^2 = 0$$

Expanding the squared term, we get:

$$-5(x^2 - 6x + 9) = 0$$

Distributing the negative five, we get:

$$-5x^2 + 30x - 45 = 0$$

Dividing both sides by $-5$, we get:

$$x^2 - 6x + 9 = 0$$

This is a quadratic equation in the form $ax^2 + bx + c = 0$. We can solve this equation using the quadratic formula:

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

In this case, $a = 1$, $b = -6$, and $c = 9$. Plugging these values into the quadratic formula, we get:

$$x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(9)}}{2(1)}$$

Simplifying the expression, we get:

$$x = \frac{6 \pm \sqrt{36 - 36}}{2}$$

This simplifies to:

$$x = \frac{6 \pm \sqrt{0}}{2}$$

Since the square root of zero is zero, we have:

$$x = \frac{6}{2}$$

Simplifying, we get:

$$x = 3$$

Therefore, the $x$-intercept of the function $g(x) = -5(x-3)^2$ is the point $(3, 0)$.

The Vertex of a Quadratic Function

The vertex of a quadratic function is the point where the function has a maximum or minimum value. It is also the point where the function changes from decreasing to increasing or from increasing to decreasing. To find the vertex of a quadratic function, we need to use the formula:

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

Finding the Vertex of $g(x)$

To find the vertex of $g(x) = -5(x-3)^2$, we need to use the formula:

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

In this case, $a = -5$ and $b = 0$. Plugging these values into the formula, we get:

$$x = -\frac{0}{2(-5)}$$

Simplifying, we get:

$$x = 0$$

Therefore, the $x$-coordinate of the vertex is $0$. To find the $y$-coordinate of the vertex, we need to plug $x = 0$ into the function:

$$g(0) = -5(0-3)^2$$

Simplifying, we get:

$$g(0) = -5(-3)^2$$

This simplifies to:

$$g(0) = -5(9)$$

Simplifying further, we get:

$$g(0) = -45$$

Therefore, the vertex of the function $g(x) = -5(x-3)^2$ is the point $(0, -45)$.

Conclusion

In this article, we have found the $x$-intercept and vertex of the quadratic function $g(x) = -5(x-3)^2$. The $x$-intercept is the point $(3, 0)$, and the vertex is the point $(0, -45)$. These key features of the function provide valuable information about its behavior and can be used to graph the function and analyze its properties.

Key Takeaways

• The $x$-intercept of a quadratic function is the point where the function intersects the $x$-axis.
• The vertex of a quadratic function is the point where the function has a maximum or minimum value.
• To find the $x$-intercept of a quadratic function, we need to set the function equal to zero and solve for $x$.
• To find the vertex of a quadratic function, we need to use the formula $x = -\frac{b}{2a}$.

Further Reading

For more information on quadratic functions and their key features, we recommend the following resources:

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

References

• [1] Larson, R. (2014). College Algebra. Cengage Learning.
• [2] Sullivan, M. (2015). Algebra and Trigonometry. Pearson Education.
• [3] Anton, H. (2016). Calculus: Early Transcendentals. John Wiley & Sons.
  Quadratic Function Q&A

Understanding Quadratic Functions

Quadratic functions are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. 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, and $a$ cannot be zero.

Quadratic Function Q&A

Here are some frequently asked questions about quadratic functions:

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

A: The $x$-intercept of a quadratic function is the point where the function intersects the $x$-axis. It is the point where the function has a value of zero.

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

A: To find the $x$-intercept of a quadratic function, you need to set the function equal to zero and solve for $x$. This will give you the $x$-coordinate of the $x$-intercept.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point where the function has a maximum or minimum value. It is also the point where the function changes from decreasing to increasing or from increasing to decreasing.

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

A: To find the vertex of a quadratic function, you need to use the formula $x = -\frac{b}{2a}$. This will give you the $x$-coordinate of the vertex.

Q: What is the $y$-coordinate of the vertex?

A: To find the $y$-coordinate of the vertex, you need to plug the $x$-coordinate of the vertex into the function.

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

A: No, a quadratic function can only have one $x$-intercept.

Q: Can a quadratic function have more than one vertex?

A: No, a quadratic function can only have one vertex.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you need to plot the $x$-intercept and the vertex, and then draw a smooth curve through the points.

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. It is the line that divides the graph of the function into two equal halves.

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 use the formula $x = -\frac{b}{2a}$. This will give you the $x$-coordinate of the axis of symmetry.

Q: What is the standard form of a quadratic function?

A: The standard form of a quadratic function is $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $a$ cannot be zero.

Q: How do I convert a quadratic function from general form to standard form?

A: To convert a quadratic function from general form to standard form, you need to group the terms and factor out the greatest common factor.

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

A: The vertex form of a quadratic function is $f(x) = a(x-h)^2 + k$, where $(h, k)$ is the vertex of the function.

Q: How do I convert a quadratic function from standard form to vertex form?

A: To convert a quadratic function from standard form to vertex form, you need to complete the square.

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

A: A quadratic function is a polynomial function of degree two, while a linear function is a polynomial function of degree one.

Q: Can a quadratic function be a linear function?

A: No, a quadratic function cannot be a linear function.

Q: Can a linear function be a quadratic function?

A: No, a linear function cannot be a quadratic function.

Conclusion

In this article, we have answered some frequently asked questions about quadratic functions. We have covered topics such as the $x$-intercept, vertex, axis of symmetry, standard form, and vertex form of a quadratic function. We hope that this article has provided you with a better understanding of quadratic functions and their key features.

Key Takeaways

• The $x$-intercept of a quadratic function is the point where the function intersects the $x$-axis.
• The vertex of a quadratic function is the point where the function has a maximum or minimum value.
• The axis of symmetry of a quadratic function is the vertical line that passes through the vertex.
• The standard form of a quadratic function is $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $a$ cannot be zero.
• The vertex form of a quadratic function is $f(x) = a(x-h)^2 + k$, where $(h, k)$ is the vertex of the function.

Further Reading

For more information on quadratic functions and their key features, we recommend the following resources:

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

References

• [1] Larson, R. (2014). College Algebra. Cengage Learning.
• [2] Sullivan, M. (2015). Algebra and Trigonometry. Pearson Education.
• [3] Anton, H. (2016). Calculus: Early Transcendentals. John Wiley & Sons.