Plot The Vertex And The Axis Of Symmetry Of This Function:$ F(x) = (x - 3)^2 + 5 $Use The Drawing Tools To Form The Correct Answers On The Graph.- Vertex: The Vertex Is The Point (3, 5).- Axis Of Symmetry: The Axis Of Symmetry Is The Vertical

Introduction

In mathematics, particularly in algebra and geometry, understanding the properties of quadratic functions is crucial. One of the key aspects of quadratic functions is the concept of the vertex and axis of symmetry. In this article, we will explore how to plot the vertex and axis of symmetry of a given quadratic function, specifically the function $f(x) = (x - 3)^2 + 5$.

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree two, which means the highest power of the variable (in this case, $x$) 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.

The Vertex of a Quadratic Function

The vertex of a quadratic function is the highest or lowest point on the graph of the function. It is the point where the parabola changes direction, from opening upwards to opening downwards or vice versa. The vertex is denoted by the point $(h, k)$, where $h$ is the x-coordinate and $k$ is the y-coordinate.

Finding the Vertex of the Given Function

To find the vertex of the given function $f(x) = (x - 3)^2 + 5$, we need to identify the values of $h$ and $k$. In this case, the function is in the form $f(x) = (x - h)^2 + k$, where $h$ is the x-coordinate of the vertex and $k$ is the y-coordinate.

By comparing the given function with the general form, we can see that $h = 3$ and $k = 5$. Therefore, the vertex of the given function is the point $(3, 5)$.

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 function. It is a line of symmetry, meaning that if we draw a line through the vertex, the two sides of the parabola will be mirror images of each other.

Finding the Axis of Symmetry of the Given Function

To find the axis of symmetry of the given function, we need to find the equation of the vertical line that passes through the vertex. Since the vertex is the point $(3, 5)$, the equation of the axis of symmetry is $x = 3$.

Plotting the Vertex and Axis of Symmetry

To plot the vertex and axis of symmetry of the given function, we can use the drawing tools to form the correct answers on the graph. The vertex is the point $(3, 5)$, and the axis of symmetry is the vertical line $x = 3$.

Conclusion

In conclusion, plotting the vertex and axis of symmetry of a quadratic function is an essential skill in mathematics. By understanding the properties of quadratic functions and using the drawing tools to form the correct answers on the graph, we can visualize the vertex and axis of symmetry of a given function. In this article, we have explored how to plot the vertex and axis of symmetry of the function $f(x) = (x - 3)^2 + 5$.

Step-by-Step Guide to Plotting the Vertex and Axis of Symmetry

Step 1: Identify the Vertex

• Identify the values of $h$ and $k$ in the given function.
• In this case, $h = 3$ and $k = 5$.
• Therefore, the vertex of the given function is the point $(3, 5)$.

Step 2: Identify the Axis of Symmetry

• Find the equation of the vertical line that passes through the vertex.
• Since the vertex is the point $(3, 5)$, the equation of the axis of symmetry is $x = 3$.

Step 3: Plot the Vertex and Axis of Symmetry

• Use the drawing tools to form the correct answers on the graph.
• Plot the vertex as the point $(3, 5)$.
• Plot the axis of symmetry as the vertical line $x = 3$.

Step 4: Verify the Plot

• Verify that the vertex and axis of symmetry are correctly plotted on the graph.
• Check that the two sides of the parabola are mirror images of each other.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the highest or lowest point on the graph of the function. It is the point where the parabola changes direction, from opening upwards to opening downwards or vice versa.

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

A: To find the vertex of a quadratic function, you need to identify the values of $h$ and $k$ in the given function. In the general form $f(x) = ax^2 + bx + c$, the vertex is the point $(h, k)$. For the function $f(x) = (x - h)^2 + k$, the vertex is the point $(h, k)$.

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

A: The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the function. It is a line of symmetry, meaning that if we draw a line through the vertex, the two sides of the parabola will be mirror images of each other.

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 need to find the equation of the vertical line that passes through the vertex. Since the vertex is the point $(h, k)$, the equation of the axis of symmetry is $x = h$.

Q: What is the significance of the vertex and axis of symmetry in a quadratic function?

A: The vertex and axis of symmetry are important concepts in quadratic functions because they help us understand the behavior of the function. The vertex represents the maximum or minimum value of the function, while the axis of symmetry represents the line of symmetry of the parabola.

Q: How do I plot the vertex and axis of symmetry on a graph?

A: To plot the vertex and axis of symmetry on a graph, you need to use the drawing tools to form the correct answers on the graph. Plot the vertex as the point $(h, k)$ and the axis of symmetry as the vertical line $x = h$.

Q: What are some common mistakes to avoid when plotting the vertex and axis of symmetry?

A: Some common mistakes to avoid when plotting the vertex and axis of symmetry include:

• Plotting the vertex at the wrong coordinates
• Plotting the axis of symmetry at the wrong equation
• Not using the drawing tools to form the correct answers on the graph
• Not verifying the plot to ensure that the vertex and axis of symmetry are correctly plotted

Q: How can I practice plotting the vertex and axis of symmetry?

A: You can practice plotting the vertex and axis of symmetry by:

• Working through examples and exercises in your textbook or online resources
• Using graphing software or online tools to plot the vertex and axis of symmetry
• Creating your own examples and exercises to practice plotting the vertex and axis of symmetry

By following these tips and practicing regularly, you can become proficient in plotting the vertex and axis of symmetry of a quadratic function.