Which Statements Are True About The Graph Of The Function $f(x) = X^2 - 8x + 5$?Select Three Options:A. The Function In Vertex Form Is $f(x) = (x-4)^2 - 11$.B. The Vertex Of The Function Is $(-8, 5$\].C. The Axis Of Symmetry

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Introduction

When analyzing the graph of a quadratic function, it's essential to understand its key characteristics, such as the vertex, axis of symmetry, and the direction of the parabola's opening. In this article, we will delve into the graph of the function f(x)=x2−8x+5f(x) = x^2 - 8x + 5 and examine three statements about its properties.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by f(x)=a(x−h)2+kf(x) = a(x-h)^2 + k, where (h,k)(h,k) represents the coordinates of the vertex. To convert the given function f(x)=x2−8x+5f(x) = x^2 - 8x + 5 into vertex form, we need to complete the square.

Completing the Square

To complete the square, we start by factoring out the coefficient of the x2x^2 term, which is 1 in this case. We then take half of the coefficient of the xx term, square it, and add it to both sides of the equation.

f(x) = x^2 - 8x + 5
= (x^2 - 8x) + 5
= (x^2 - 8x + 16) - 16 + 5
= (x - 4)^2 - 11

Vertex Form of the Function

From the previous section, we have successfully converted the function f(x)=x2−8x+5f(x) = x^2 - 8x + 5 into vertex form: f(x)=(x−4)2−11f(x) = (x-4)^2 - 11. This confirms that statement A is true.

Vertex of the Function

The vertex of the function is given by the coordinates (h,k)(h,k) in the vertex form. In this case, the vertex form is f(x)=(x−4)2−11f(x) = (x-4)^2 - 11, so the vertex is (4,−11)(4, -11). This means that statement B is false.

Axis of Symmetry

The axis of symmetry of a quadratic function is a vertical line that passes through the vertex. In this case, the axis of symmetry is the line x=4x = 4. This confirms that the axis of symmetry is indeed x=4x = 4.

Conclusion

In conclusion, we have analyzed the graph of the function f(x)=x2−8x+5f(x) = x^2 - 8x + 5 and examined three statements about its properties. We have confirmed that statement A is true, statement B is false, and statement C is true.

Key Takeaways

  • The vertex form of a quadratic function is given by f(x)=a(x−h)2+kf(x) = a(x-h)^2 + k, where (h,k)(h,k) represents the coordinates of the vertex.
  • To convert a quadratic function into vertex form, we need to complete the square.
  • The axis of symmetry of a quadratic function is a vertical line that passes through the vertex.

Final Answer

The final answer is:

  • Statement A is true: The function in vertex form is f(x)=(x−4)2−11f(x) = (x-4)^2 - 11.
  • Statement B is false: The vertex of the function is (−8,5)(-8, 5).
  • Statement C is true: The axis of symmetry is x=4x = 4.

Introduction

In our previous article, we analyzed the graph of the function f(x)=x2−8x+5f(x) = x^2 - 8x + 5 and examined three statements about its properties. In this article, we will address some frequently asked questions about the graph of this function.

Q: What is the vertex of the function?

A: The vertex of the function is given by the coordinates (h,k)(h,k) in the vertex form. In this case, the vertex form is f(x)=(x−4)2−11f(x) = (x-4)^2 - 11, so the vertex is (4,−11)(4, -11).

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

A: The axis of symmetry of a quadratic function is a vertical line that passes through the vertex. In this case, the axis of symmetry is the line x=4x = 4.

Q: How do I convert the function f(x)=x2−8x+5f(x) = x^2 - 8x + 5 into vertex form?

A: To convert the function into vertex form, we need to complete the square. We start by factoring out the coefficient of the x2x^2 term, which is 1 in this case. We then take half of the coefficient of the xx term, square it, and add it to both sides of the equation.

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

A: The vertex form of a quadratic function is useful because it allows us to easily identify the vertex and the axis of symmetry of the function. This can be helpful in graphing the function and understanding its behavior.

Q: Can you provide an example of how to use the vertex form to graph a quadratic function?

A: Yes, let's consider the function f(x)=(x−4)2−11f(x) = (x-4)^2 - 11. To graph this function, we can start by plotting the vertex (4,−11)(4, -11). We can then use the axis of symmetry x=4x = 4 to help us plot the rest of the graph.

Q: How do I determine the direction of the parabola's opening?

A: The direction of the parabola's opening can be determined by the sign of the coefficient of the x2x^2 term. If the coefficient is positive, the parabola opens upward. If the coefficient is negative, the parabola opens downward.

Q: Can you provide an example of how to determine the direction of the parabola's opening?

A: Yes, let's consider the function f(x)=x2−8x+5f(x) = x^2 - 8x + 5. The coefficient of the x2x^2 term is 1, which is positive. Therefore, the parabola opens upward.

Conclusion

In conclusion, we have addressed some frequently asked questions about the graph of the function f(x)=x2−8x+5f(x) = x^2 - 8x + 5. We hope that this article has been helpful in providing a better understanding of the graph of this function.

Key Takeaways

  • The vertex form of a quadratic function is given by f(x)=a(x−h)2+kf(x) = a(x-h)^2 + k, where (h,k)(h,k) represents the coordinates of the vertex.
  • To convert a quadratic function into vertex form, we need to complete the square.
  • The axis of symmetry of a quadratic function is a vertical line that passes through the vertex.
  • The direction of the parabola's opening can be determined by the sign of the coefficient of the x2x^2 term.

Final Answer

The final answer is:

  • The vertex of the function is (4,−11)(4, -11).
  • The axis of symmetry of the function is x=4x = 4.
  • The direction of the parabola's opening is upward.