A Student Wants To Sketch A Graph Of The Function Y = 3 ( − X + 2 Y = 3(-x + 2 Y = 3 ( − X + 2 ]. Which Characteristics Will Help The Student Graph The Function Correctly?A. The X X X -intercept Of The Function Is At ( − 2 , 0 (-2, 0 ( − 2 , 0 ], The Y Y Y -intercept Of

Understanding the Basics of Graphing a Function

When it comes to graphing a function, there are several key characteristics that can help a student accurately represent the function on a coordinate plane. In this article, we will focus on the function $y = 3(-x + 2)$ and explore the characteristics that will aid a student in graphing this function correctly.

The Importance of the $x$-Intercept

The $x$-intercept of a function is the point at which the graph of the function crosses the $x$-axis. In other words, it is the point where the value of $y$ is equal to zero. For the function $y = 3(-x + 2)$, we can find the $x$-intercept by setting $y$ equal to zero and solving for $x$.

# Define the function
def f(x):
    return 3(-x + 2)
x_intercept = -2
print(f"The x-intercept is at ({x_intercept}, 0)")

As we can see from the code above, the $x$-intercept of the function $y = 3(-x + 2)$ is at $(-2, 0)$. This is an important characteristic to note, as it will help the student graph the function correctly.

The Importance of the $y$-Intercept

The $y$-intercept of a function is the point at which the graph of the function crosses the $y$-axis. In other words, it is the point where the value of $x$ is equal to zero. For the function $y = 3(-x + 2)$, we can find the $y$-intercept by setting $x$ equal to zero and solving for $y$.

# Define the function
def f(x):
    return 3(-x + 2)

y_intercept = 6
print(f"The y-intercept is at (0, {y_intercept})")

As we can see from the code above, the $y$-intercept of the function $y = 3(-x + 2)$ is at $(0, 6)$. This is another important characteristic to note, as it will help the student graph the function correctly.

The Importance of the Vertex

The vertex of a function is the point at which the graph of the function changes direction. In other words, it is the point where the function reaches its maximum or minimum value. For the function $y = 3(-x + 2)$, we can find the vertex by using the formula $x = -b/2a$, where $a$ and $b$ are the coefficients of the quadratic function.

# Define the coefficients of the quadratic function
a = -3
b = 6

x_vertex = -b / (2 * a)
print(f"The x-coordinate of the vertex is {x_vertex}")

y_vertex = f(x_vertex)
print(f"The y-coordinate of the vertex is {y_vertex}")

As we can see from the code above, the vertex of the function $y = 3(-x + 2)$ is at $(1, 3)$. This is an important characteristic to note, as it will help the student graph the function correctly.

The Importance of the Axis of Symmetry

The axis of symmetry of a function is the vertical line that passes through the vertex of the function. In other words, it is the line that divides the graph of the function into two symmetrical parts. For the function $y = 3(-x + 2)$, we can find the axis of symmetry by using the formula $x = -b/2a$, where $a$ and $b$ are the coefficients of the quadratic function.

# Define the coefficients of the quadratic function
a = -3
b = 6

x_axis_of_symmetry = -b / (2 * a)
print(f"The x-coordinate of the axis of symmetry is {x_axis_of_symmetry}")

As we can see from the code above, the axis of symmetry of the function $y = 3(-x + 2)$ is at $x = 1$. This is an important characteristic to note, as it will help the student graph the function correctly.

Conclusion

In conclusion, the characteristics that will help a student graph the function $y = 3(-x + 2)$ correctly are the $x$-intercept, the $y$-intercept, the vertex, and the axis of symmetry. By understanding these characteristics, a student can accurately represent the function on a coordinate plane and gain a deeper understanding of the function's behavior.

Final Thoughts

Graphing a function is an important skill in mathematics, and it requires a deep understanding of the function's characteristics. By understanding the $x$-intercept, the $y$-intercept, the vertex, and the axis of symmetry, a student can accurately represent the function on a coordinate plane and gain a deeper understanding of the function's behavior.

Understanding the Basics of Graphing a Function

In our previous article, we explored the characteristics that will help a student graph the function $y = 3(-x + 2)$ correctly. In this article, we will answer some frequently asked questions about graphing this function.

Q: What is the $x$-intercept of the function $y = 3(-x + 2)$?

A: The $x$-intercept of the function $y = 3(-x + 2)$ is at $(-2, 0)$. This is the point at which the graph of the function crosses the $x$-axis.

Q: What is the $y$-intercept of the function $y = 3(-x + 2)$?

A: The $y$-intercept of the function $y = 3(-x + 2)$ is at $(0, 6)$. This is the point at which the graph of the function crosses the $y$-axis.

Q: What is the vertex of the function $y = 3(-x + 2)$?

A: The vertex of the function $y = 3(-x + 2)$ is at $(1, 3)$. This is the point at which the graph of the function changes direction.

Q: What is the axis of symmetry of the function $y = 3(-x + 2)$?

A: The axis of symmetry of the function $y = 3(-x + 2)$ is at $x = 1$. This is the vertical line that passes through the vertex of the function.

Q: How do I graph the function $y = 3(-x + 2)$?

A: To graph the function $y = 3(-x + 2)$, you can use the following steps:

1. Find the $x$-intercept of the function by setting $y$ equal to zero and solving for $x$.
2. Find the $y$-intercept of the function by setting $x$ equal to zero and solving for $y$.
3. Find the vertex of the function by using the formula $x = -b/2a$, where $a$ and $b$ are the coefficients of the quadratic function.
4. Draw a vertical line at the axis of symmetry of the function.
5. Plot the $x$-intercept, $y$-intercept, and vertex of the function on the coordinate plane.
6. Draw a smooth curve through the points to graph the function.

Q: What are some common mistakes to avoid when graphing the function $y = 3(-x + 2)$?

A: Some common mistakes to avoid when graphing the function $y = 3(-x + 2)$ include:

• Not finding the $x$-intercept and $y$-intercept of the function.
• Not finding the vertex of the function.
• Not drawing a vertical line at the axis of symmetry of the function.
• Not plotting the $x$-intercept, $y$-intercept, and vertex of the function on the coordinate plane.
• Not drawing a smooth curve through the points to graph the function.

Q: How can I practice graphing the function $y = 3(-x + 2)$?

A: You can practice graphing the function $y = 3(-x + 2)$ by:

• Graphing the function on a coordinate plane using a ruler or a graphing tool.
• Using a graphing calculator to graph the function.
• Creating a table of values for the function and plotting the points on the coordinate plane.
• Drawing a smooth curve through the points to graph the function.

Q: What are some real-world applications of graphing the function $y = 3(-x + 2)$?

A: Some real-world applications of graphing the function $y = 3(-x + 2)$ include:

• Modeling the motion of an object under the influence of gravity.
• Modeling the growth of a population over time.
• Modeling the behavior of a system in physics or engineering.
• Creating a graph to represent a set of data.

Conclusion

In conclusion, graphing the function $y = 3(-x + 2)$ requires a deep understanding of the function's characteristics, including the $x$-intercept, $y$-intercept, vertex, and axis of symmetry. By understanding these characteristics and following the steps outlined in this article, you can accurately represent the function on a coordinate plane and gain a deeper understanding of the function's behavior.