Which Describes The Graph Of Y = ( X − 4 ) 2 − 1 Y=(x-4)^2-1 Y = ( X − 4 ) 2 − 1 ?A. Opens Up With A Vertex At ( 4 , − 1 (4,-1 ( 4 , − 1 ] B. Opens Up With A Vertex At ( − 4 , − 1 (-4,-1 ( − 4 , − 1 ] C. Opens Down With A Vertex At ( 4 , − 1 (4,-1 ( 4 , − 1 ] D. Opens Down With A Vertex At ( − 4 , − 1 (-4,-1 ( − 4 , − 1 ]

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 $f(x) = ax^2 + bx + c$, 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 quadratic function $y = (x - 4)^2 - 1$ and determine which of the given options describes it.

The Standard Form of a Quadratic Function

The standard form of a quadratic function is $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. In this form, the vertex is the lowest or highest point of the parabola, depending on whether $a$ is positive or negative. If $a$ is positive, the parabola opens up, and if $a$ is negative, the parabola opens down.

The Given Quadratic Function

The given quadratic function is $y = (x - 4)^2 - 1$. To determine the vertex of this parabola, we need to expand the squared term and simplify the expression.

Expanding the Squared Term

Using the formula $(x - a)^2 = x^2 - 2ax + a^2$, we can expand the squared term in the given quadratic function as follows:

$$(x - 4)^2 = x^2 - 2(4)x + 4^2 = x^2 - 8x + 16$$

Simplifying the Expression

Now, we can substitute the expanded squared term back into the original quadratic function:

$$y = (x - 4)^2 - 1 = x^2 - 8x + 16 - 1 = x^2 - 8x + 15$$

Determining the Vertex

The vertex of the parabola is the point where the parabola changes direction. To find the vertex, we need to complete the square or use the formula $x = -\frac{b}{2a}$.

Using the Formula

Using the formula $x = -\frac{b}{2a}$, we can find the x-coordinate of the vertex as follows:

$$x = -\frac{b}{2a} = -\frac{-8}{2(1)} = 4$$

Finding the y-Coordinate

Now that we have the x-coordinate of the vertex, we can find the y-coordinate by substituting $x = 4$ into the simplified expression:

$$y = (4)^2 - 8(4) + 15 = 16 - 32 + 15 = -1$$

Conclusion

Therefore, the vertex of the parabola is $(4, -1)$. Since the coefficient of the squared term is positive, the parabola opens up.

Which Describes the Graph of $y = (x - 4)^2 - 1$?

Based on our analysis, we can conclude that the graph of $y = (x - 4)^2 - 1$ is a parabola that opens up with a vertex at $(4, -1)$.

Answer

The correct answer is:

A. Opens up with a vertex at $(4, -1)$

Discussion

This problem requires a good understanding of quadratic functions and their graphs. The student needs to be able to identify the vertex of the parabola and determine whether the parabola opens up or down. This problem is a good example of how to use the standard form of a quadratic function to determine the vertex and the direction of the parabola.

Key Takeaways

• The standard form of a quadratic function is $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.
• The vertex of a parabola is the point where the parabola changes direction.
• The parabola opens up if the coefficient of the squared term is positive and opens down if the coefficient is negative.
• The graph of a quadratic function can be determined by identifying the vertex and the direction of the parabola.
  Quadratic Function Graphs: A Q&A Guide =============================================

Introduction

In our previous article, we discussed the graph of the quadratic function $y = (x - 4)^2 - 1$ and determined that it is a parabola that opens up with a vertex at $(4, -1)$. In this article, we will provide a Q&A guide to help you better understand quadratic function graphs.

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 $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

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

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

Q: How do I determine the vertex of a parabola?

A: To determine the vertex of a parabola, you can use the formula $x = -\frac{b}{2a}$ to find the x-coordinate of the vertex, and then substitute this value into the function to find the y-coordinate.

Q: What is the significance of the vertex of a parabola?

A: The vertex of a parabola is the point where the parabola changes direction. It is also the minimum or maximum point of the parabola, depending on whether the parabola opens up or down.

Q: How do I determine whether a parabola opens up or down?

A: To determine whether a parabola opens up or down, you need to look at the coefficient of the squared term. If the coefficient is positive, the parabola opens up, and if the coefficient is negative, the parabola opens down.

Q: What is the difference between a parabola that opens up and one that opens down?

A: A parabola that opens up has a minimum point, while a parabola that opens down has a maximum point. The vertex of a parabola that opens up is the minimum point, and the vertex of a parabola that opens down is the maximum point.

Q: How do I graph a quadratic function?

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

1. Determine the vertex of the parabola.
2. Determine whether the parabola opens up or down.
3. Plot the vertex on the coordinate plane.
4. Plot two points on either side of the vertex, one above and one below.
5. Draw a smooth curve through the points to form the parabola.

Q: What are some common types of quadratic functions?

A: Some common types of quadratic functions include:

• Linear quadratic functions: $f(x) = ax + b$
• Quadratic functions with a positive leading coefficient: $f(x) = ax^2 + bx + c$
• Quadratic functions with a negative leading coefficient: $f(x) = -ax^2 + bx + c$

Q: How do I use quadratic functions in real-world applications?

A: Quadratic functions have many real-world applications, including:

• Modeling the trajectory of a projectile
• Determining the maximum or minimum value of a function
• Finding the vertex of a parabola
• Graphing quadratic functions

Conclusion

In this article, we provided a Q&A guide to help you better understand quadratic function graphs. We discussed the standard form of a quadratic function, how to determine the vertex of a parabola, and how to graph a quadratic function. We also covered some common types of quadratic functions and their real-world applications.