Which Statements Are True About The Graph Of The Function F ( X ) = X 2 − 8 X + 5 F(x) = X^2 - 8x + 5 F ( X ) = X 2 − 8 X + 5 ? Select Three Options.A. The Function In Vertex Form Is F ( X ) = ( X − 4 ) 2 − 11 F(x) = (x-4)^2 - 11 F ( X ) = ( X − 4 ) 2 − 11 .B. The Vertex Of The Function Is ( − 8 , 5 (-8, 5 ( − 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'll delve into the graph of the function $f(x) = x^2 - 8x + 5$ and examine three statements about its properties.

Vertex Form of a Quadratic Function

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

Completing the Square

To complete the square, we'll start by factoring out the coefficient of the $x^2$ term, which is 1 in this case.

$$f(x) = x^2 - 8x + 5$$

Next, we'll add and subtract the square of half the coefficient of the $x$ term, which is $(-8/2)^2 = 16$. This will allow us to create a perfect square trinomial.

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

Now, we can rewrite the expression as a perfect square trinomial.

$$f(x) = (x - 4)^2 - 11$$

Comparing with the Given Options

Let's compare the vertex form we obtained with the given options.

• Option A: The function in vertex form is $f(x) = (x-4)^2 - 11$. This statement is TRUE.
• Option B: The vertex of the function is $(-8, 5)$. This statement is FALSE, as the vertex is actually at $(4, -11)$.
• Option C: The axis of symmetry is $x = 4$. This statement is TRUE, as the axis of symmetry is indeed $x = 4$.

Understanding the Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola. In this case, the axis of symmetry is $x = 4$, which means that the parabola is symmetric about the line $x = 4$.

Understanding the Vertex

The vertex of the parabola is the point where the parabola changes direction. In this case, the vertex is at $(4, -11)$, which means that the parabola opens upward.

Conclusion

In conclusion, the three statements about the graph of the function $f(x) = x^2 - 8x + 5$ are:

• The function in vertex form is $f(x) = (x-4)^2 - 11$.
• The axis of symmetry is $x = 4$.
• The vertex of the function is $(4, -11)$.

These statements are all TRUE, and they provide valuable information about the graph of the function.

Key Takeaways

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

Final Thoughts

In this article, we've explored the graph of the function $f(x) = x^2 - 8x + 5$ and examined three statements about its properties. We've seen that the function can be converted into vertex form, and we've identified the axis of symmetry and the vertex of the parabola. These concepts are essential for understanding the graph of a quadratic function, and they have numerous applications in mathematics and other fields.

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Introduction

In our previous article, we explored the graph of the function $f(x) = x^2 - 8x + 5$ and examined three statements about its properties. In this article, we'll answer some frequently asked questions about the graph of this function.

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

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

Q: How do I convert a quadratic function into vertex form?

A: To convert a quadratic function into vertex form, you need to complete the square. This involves adding and subtracting the square of half the coefficient of the $x$ term.

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

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

Q: What is the vertex of the parabola?

A: The vertex of the parabola is the point where the parabola changes direction. In this case, the vertex is at $(4, -11)$.

Q: Is the parabola opening upward or downward?

A: The parabola is opening upward, since the coefficient of the $x^2$ term is positive.

Q: How do I determine the direction of the parabola?

A: To determine the direction of the parabola, you need to look at the coefficient of the $x^2$ term. If it's positive, the parabola opens upward. If it's negative, the parabola opens downward.

Q: Can I use the vertex form to graph the parabola?

A: Yes, you can use the vertex form to graph the parabola. Simply plot the vertex and then use the axis of symmetry to draw the parabola.

Q: What are some real-world applications of quadratic functions?

A: Quadratic functions have numerous real-world applications, including physics, engineering, economics, and computer science. They're used to model the motion of objects, the growth of populations, and the behavior of electrical circuits, among other things.

Q: How do I find the x-intercepts of the parabola?

A: To find the x-intercepts of the parabola, you need to set the function equal to zero and solve for $x$. This will give you the x-coordinates of the x-intercepts.

Q: Can I use the axis of symmetry to find the x-intercepts?

A: Yes, you can use the axis of symmetry to find the x-intercepts. Since the axis of symmetry is $x = 4$, you know that the x-intercepts will be equidistant from the axis of symmetry.

Conclusion

In this article, we've answered some frequently asked questions about the graph of the function $f(x) = x^2 - 8x + 5$. We've covered topics such as the vertex form of a quadratic function, converting a quadratic function into vertex form, and the of symmetry and vertex of the parabola. These concepts are essential for understanding the graph of a quadratic function, and they have numerous applications in mathematics and other fields.

Key Takeaways

• The vertex form of a quadratic function is given by $f(x) = a(x-h)^2 + k$, where $(h,k)$ represents the coordinates of the vertex.
• To convert a quadratic function into vertex form, you need to complete the square.
• The axis of symmetry is a vertical line that passes through the vertex of the parabola.
• The vertex of the parabola is the point where the parabola changes direction.
• The parabola is opening upward, since the coefficient of the $x^2$ term is positive.
• You can use the vertex form to graph the parabola.
• Quadratic functions have numerous real-world applications.

Final Thoughts

In this article, we've explored some frequently asked questions about the graph of the function $f(x) = x^2 - 8x + 5$. We've covered topics such as the vertex form of a quadratic function, converting a quadratic function into vertex form, and the axis of symmetry and vertex of the parabola. These concepts are essential for understanding the graph of a quadratic function, and they have numerous applications in mathematics and other fields.