Part Of The Graph Of The Function F ( X ) = ( X − 1 ) ( X + 7 F(x)=(x-1)(x+7 F ( X ) = ( X − 1 ) ( X + 7 ] Is Shown Below.Which Statements About The Function Are True? Select Three Options.A. The Vertex Of The Function Is At ( − 4 , − 15 (-4,-15 ( − 4 , − 15 ].B. The Vertex Of The Function Is At

Introduction

In mathematics, a quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The graph of a quadratic function is a parabola, and it can be represented in various forms, including factored form, standard form, and vertex form. In this article, we will discuss the graph of the function $f(x) = (x-1)(x+7)$ and determine which statements about the function are true.

The Graph of the Function

The graph of the function $f(x) = (x-1)(x+7)$ is shown below.

Analyzing the Graph

From the graph, we can see that the function has a minimum point, which is the vertex of the parabola. The vertex is the lowest or highest point on the graph, and it is represented by the point $(h, k)$, where $h$ is the x-coordinate and $k$ is the y-coordinate.

Statement A: The Vertex of the Function is at $(-4,-15)$

To determine if statement A is true, we need to find the vertex of the function. The vertex form of a quadratic function is given by $f(x) = a(x-h)^2 + k$, where $(h, k)$ is the vertex. We can rewrite the given function in vertex form by completing the square.

import sympy as sp
x = sp.symbols('x')
f = (x-1)*(x+7)
f_vertex = sp.expand(f)
print(f_vertex)

The output of the code is:

x**2 + 6*x - 7

Now, we can complete the square to find the vertex.

import sympy as sp
x = sp.symbols('x')
f = x**2 + 6*x - 7
f_completed_square = sp.expand((x + 3)**2 - 16)
print(f_completed_square)

The output of the code is:

x**2 + 6*x - 7

Comparing the two expressions, we can see that the vertex form of the function is $f(x) = (x+3)^2 - 16$. Therefore, the vertex of the function is at $(-3, -16)$, not $(-4, -15)$.

Conclusion

In conclusion, statement A is false. The vertex of the function is at $(-3, -16)$, not $(-4, -15)$.

Statement B: The Vertex of the Function is at $(-3, -16)$

From the previous analysis, we know that the vertex of the function is at $(-3, -16)$. Therefore, statement B is true.

Statement C: The Function has a Maximum Point

From the graph, we can see that the function has a minimum point, which is the vertex of the parabola. Therefore, the function does not have a maximum point.

Conclusion

In conclusion, statement C is false. The function has a minimum point, but it does not have a maximum point.

Final Answer

The final answer is:

• Statement A is false.
• Statement B is true.
• Statement C is false.

Introduction

In our previous article, we discussed the graph of the function $f(x) = (x-1)(x+7)$ and analyzed its properties. In this article, we will answer some frequently asked questions about quadratic functions.

Q: What is a quadratic function?

A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. It can be represented in various forms, including factored form, standard form, and vertex form.

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

The standard form of a quadratic function is given by $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

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

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

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

To find the vertex of a quadratic function, you can use the formula $h = -\frac{b}{2a}$ and $k = f(h)$. Alternatively, you can complete the square to find the vertex.

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

The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the parabola. It is given by the equation $x = h$.

A: How do I find the axis of symmetry of a quadratic function?

To find the axis of symmetry of a quadratic function, you can use the formula $x = h$, where $h$ is the x-coordinate of the vertex.

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

The x-intercept of a quadratic function is the point where the graph of the function intersects the x-axis. It is given by the equation $f(x) = 0$.

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

To find the x-intercept of a quadratic function, you can set the function equal to zero and solve for $x$.

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

The y-intercept of a quadratic function is the point where the graph of the function intersects the y-axis. It is given by the equation $f(0)$.

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

To find the y-intercept of a quadratic function, you can substitute $x = 0$ into the function and evaluate it.

Q: How do I graph a quadratic function?

To graph a quadratic function, you can use a graphing calculator or a computer program. Alternatively, you can plot points on the graph and connect them with a smooth curve.

A: What are some common applications of quadratic functions?

Quadratic functions have many real-world applications, including:

• Modeling the trajectory of a projectile
• Finding the maximum or minimum value of a function
• Determining the area or volume of a shape
• Solving optimization problems

Conclusion

In conclusion, quadratic functions are an important concept in mathematics, and they have many real-world applications. By understanding the properties and behavior of quadratic functions, you can solve a wide range of problems and make informed decisions.

Additional Resources

For more information on quadratic functions, you can consult the following resources:

• Khan Academy: Quadratic Functions
• Mathway: Quadratic Functions
• Wolfram Alpha: Quadratic Functions

Note: The resources listed above are just a few examples of the many online resources available for learning about quadratic functions.