Consider The Function Below: F ( X ) = X 2 − 6 X + 5 F(x) = X^2 - 6x + 5 F ( X ) = X 2 − 6 X + 5 Plot The Following On The Provided Coordinate Plane:- The { X$}$-intercept(s)- The { Y$}$-intercept- The Vertex- The Axis Of Symmetry

Introduction

In mathematics, functions are used to describe the relationship between variables. A function can be represented graphically on a coordinate plane, which provides a visual representation of the function's behavior. In this article, we will consider the function $f(x) = x^2 - 6x + 5$ and plot its key features on the provided coordinate plane.

The Function

The given function is a quadratic function, which can be written in the form $f(x) = ax^2 + bx + c$. In this case, the function is $f(x) = x^2 - 6x + 5$, where $a = 1$, $b = -6$, and $c = 5$.

Finding the x-Intercept(s)

The x-intercept(s) of a function are the points where the function intersects the x-axis. To find the x-intercept(s), we need to set $y = 0$ and solve for $x$. In this case, we have:

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

We can solve this quadratic equation using the quadratic formula:

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

Substituting the values of $a$, $b$, and $c$, we get:

$$x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(5)}}{2(1)}$$

Simplifying, we get:

$$x = \frac{6 \pm \sqrt{36 - 20}}{2}$$

$$x = \frac{6 \pm \sqrt{16}}{2}$$

$$x = \frac{6 \pm 4}{2}$$

Therefore, the x-intercept(s) are:

$$x = \frac{6 + 4}{2} = 5$$

$$x = \frac{6 - 4}{2} = 1$$

Finding the y-Intercept

The y-intercept of a function is the point where the function intersects the y-axis. To find the y-intercept, we need to set $x = 0$ and solve for $y$. In this case, we have:

$$f(x) = x^2 - 6x + 5$$

Substituting $x = 0$, we get:

$$f(0) = 0^2 - 6(0) + 5$$

$$f(0) = 5$$

Therefore, the y-intercept is $(0, 5)$.

Finding the Vertex

The vertex of a quadratic function is the point where the function changes from decreasing to increasing or vice versa. To find the vertex, we need to use the formula:

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

Substituting the values of $a$ and $b$, we get:

$$x = -\frac{-6}{2(1)}$$

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

$$x = 3$$

Substituting $x = 3$ into the function, we get:

$$f(3) = 3^2 - 6(3) + 5$$

$$f(3) = 9 - 18 + 5$$

$$f(3) = -4$$

Therefore, the vertex is $(3, -4)$.

Finding the Axis of Symmetry

The axis of symmetry of a quadratic function is the vertical line that passes through the vertex. To find the axis of symmetry, we need to use the formula:

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

Substituting the values of $a$ and $b$, we get:

$$x = -\frac{-6}{2(1)}$$

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

$$x = 3$$

Therefore, the axis of symmetry is the vertical line $x = 3$.

Plotting the Key Features

Now that we have found the x-intercept(s), y-intercept, vertex, and axis of symmetry, we can plot these key features on the coordinate plane.

• The x-intercept(s) are $(5, 0)$ and $(1, 0)$.
• The y-intercept is $(0, 5)$.
• The vertex is $(3, -4)$.
• The axis of symmetry is the vertical line $x = 3$.

Conclusion

In this article, we have considered the function $f(x) = x^2 - 6x + 5$ and plotted its key features on the coordinate plane. We have found the x-intercept(s), y-intercept, vertex, and axis of symmetry, and plotted these features on the coordinate plane. This graphical representation provides a visual understanding of the function's behavior and can be used to analyze and understand the function's properties.

References

• [1] "Quadratic Functions". Math Open Reference. Retrieved 2023-02-20.
• [2] "Graphing Quadratic Functions". Purplemath. Retrieved 2023-02-20.

Glossary

• Axis of Symmetry: The vertical line that passes through the vertex of a quadratic function.
• Quadratic Function: A function of the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.
• Vertex: The point where a quadratic function changes from decreasing to increasing or vice versa.
• X-Intercept: The point where a function intersects the x-axis.
• Y-Intercept: The point where a function intersects the y-axis.
  Quadratic Function Q&A =========================

Frequently Asked Questions

In this article, we will answer some frequently asked questions about quadratic functions, including their graph, key features, and properties.

Q: What is a quadratic function?

A: A quadratic function is a function of the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. Quadratic functions are used to model a wide range of real-world phenomena, including the motion of objects, the growth of populations, and the behavior of electrical circuits.

Q: What is the graph of a quadratic function?

A: The graph of a quadratic function is a parabola, which is a U-shaped curve. The graph can be either upward-facing or downward-facing, depending on the value of $a$. If $a > 0$, the graph is upward-facing, and if $a < 0$, the graph is downward-facing.

Q: What are the key features of a quadratic function?

A: The key features of a quadratic function include:

• Vertex: The point where the function changes from decreasing to increasing or vice versa.
• Axis of Symmetry: The vertical line that passes through the vertex.
• X-Intercepts: The points where the function intersects the x-axis.
• Y-Intercept: The point where the function intersects the y-axis.

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

A: To find the vertex of a quadratic function, you can use the formula:

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

Substituting the values of $a$ and $b$, you can find the x-coordinate of the vertex. Then, substitute the x-coordinate into the function to find the y-coordinate.

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 can use the formula:

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

This will give you the x-coordinate of the axis of symmetry.

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

A: To find the x-intercepts of a quadratic function, you can set $y = 0$ and solve for $x$. This will give you the x-coordinates of the x-intercepts.

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

A: To find the y-intercept of a quadratic function, you can set $x = 0$ and solve for $y$. This will give you the y-coordinate of the y-intercept.

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

A: A quadratic function is a function of the form $f(x) = ax^2 + bx + c$, while a linear function is a function of the form $f(x) = mx + b$. Quadratic functions have a parabolic graph, while linear functions have a straight-line graph.

Q: Can quadratic functions be used to model real-world phenomena?

A: Yes, quadratic functions can be used to model a wide range of real-world phenomena, including the motion of objects, the growth of populations, and the behavior of electrical circuits.

Q: How do I graph a quadratic function?

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

1. Find the vertex of the function.
2. Find the axis of symmetry of the function.
3. Find the x-intercepts of the function.
4. Find the y-intercept of the function.
5. Plot the points on a coordinate plane.

Conclusion

In this article, we have answered some frequently asked questions about quadratic functions, including their graph, key features, and properties. We have also provided examples and explanations to help you understand the concepts. By following the steps outlined in this article, you can graph a quadratic function and analyze its behavior.

References

• [1] "Quadratic Functions". Math Open Reference. Retrieved 2023-02-20.
• [2] "Graphing Quadratic Functions". Purplemath. Retrieved 2023-02-20.

Glossary

• Axis of Symmetry: The vertical line that passes through the vertex of a quadratic function.
• Linear Function: A function of the form $f(x) = mx + b$.
• Parabola: A U-shaped curve that is the graph of a quadratic function.
• Quadratic Function: A function of the form $f(x) = ax^2 + bx + c$.
• Vertex: The point where a quadratic function changes from decreasing to increasing or vice versa.
• X-Intercept: The point where a function intersects the x-axis.
• Y-Intercept: The point where a function intersects the y-axis.