Graph The Function $f(x) = -6x^2 + 7$.1. Plot The Vertex Of The Parabola.2. Plot Another Point On The Parabola. (Note: If You Make A Mistake, You Can Erase Your Parabola By Selecting The Second Point And Placing It On Top Of The First.)

Feb 25, 2025 by ADMIN 237 views

Graphing the Function $f(x) = -6x^2 + 7$

Understanding the Function

The given function is a quadratic function in the form of $f(x) = ax^2 + bx + c$ , where $a = -6$ , $b = 0$ , and $c = 7$ . The coefficient of the squared term, $a$ , determines the direction and width of the parabola. Since $a$ is negative, the parabola opens downwards, indicating that the vertex is the maximum point of the function.

Finding the Vertex

To find the vertex of the parabola, we can use the formula $x = -\frac{b}{2a}$ . In this case, $b = 0$ , so the formula simplifies to $x = 0$ . This means that the vertex of the parabola is located at the point $(0, f(0))$ . To find the y-coordinate of the vertex, we substitute $x = 0$ into the function:

$f(0) = -6(0)^2 + 7 = 7$

Therefore, the vertex of the parabola is located at the point $(0, 7)$ .

Plotting the Vertex

To plot the vertex, we need to locate the point $(0, 7)$ on the coordinate plane. We can do this by drawing a vertical line at $x = 0$ and a horizontal line at $y = 7$ . The point of intersection of these two lines is the vertex of the parabola.

Plotting Another Point

To plot another point on the parabola, we need to choose a value of $x$ and substitute it into the function to find the corresponding value of $y$ . Let's choose $x = 2$ as our test point. Substituting $x = 2$ into the function, we get:

$f(2) = -6(2)^2 + 7 = -24 + 7 = -17$

Therefore, the point $(2, -17)$ is located on the parabola.

Plotting the Point

To plot the point $(2, -17)$ , we need to locate the point on the coordinate plane. We can do this by drawing a vertical line at $x = 2$ and a horizontal line at $y = -17$ . The point of intersection of these two lines is the point $(2, -17)$ .

Graphing the Parabola

To graph the parabola, we need to plot several points on the parabola and connect them with a smooth curve. We can use the vertex and the point $(2, -17)$ as two of the points to graph the parabola.

Graphing the Parabola

Here is a step-by-step guide to graphing the parabola:

Plot the vertex at the point $(0, 7)$ .
Plot the point $(2, -17)$ .
Draw a smooth curve through the two points to graph the parabola.

Graph of the Parabola

Here is the graph of the parabola:

Graph of the Parabola

The graph of the parabola is a downward-facing parabola with its vertex at the point $(0, 7)$ . The parabola opens downwards, indicating that the vertex is the maximum point of the function.

Key Features of the Graph

The graph of the parabola has several key features:

Vertex: The vertex of the parabola is located at the point $(0, 7)$ .
Axis of Symmetry: The axis of symmetry of the parabola is the vertical line $x = 0$ .
Direction: The parabola opens downwards, indicating that the vertex is the maximum point of the function.
Width: The width of the parabola is determined by the coefficient of the squared term, $a = -6$ .

Conclusion

In this article, we graphed the function $f(x) = -6x^2 + 7$ and identified its key features. We found the vertex of the parabola, plotted another point on the parabola, and graphed the parabola. The graph of the parabola is a downward-facing parabola with its vertex at the point $(0, 7)$ .
Graphing the Function $f(x) = -6x^2 + 7$ : Q&A

Understanding the Function

Q&A

Q: What is the vertex of the parabola? A: The vertex of the parabola is located at the point $(0, 7)$ .

Q: How do I find the vertex of the parabola? A: To find the vertex of the parabola, you can use the formula $x = -\frac{b}{2a}$ . In this case, $b = 0$ , so the formula simplifies to $x = 0$ . This means that the vertex of the parabola is located at the point $(0, f(0))$ . To find the y-coordinate of the vertex, you substitute $x = 0$ into the function.

Q: What is the axis of symmetry of the parabola? A: The axis of symmetry of the parabola is the vertical line $x = 0$ .

Q: How do I graph the parabola? A: To graph the parabola, you need to plot several points on the parabola and connect them with a smooth curve. You can use the vertex and the point $(2, -17)$ as two of the points to graph the parabola.

Q: What are the key features of the graph of the parabola? A: The graph of the parabola has several key features:

Vertex: The vertex of the parabola is located at the point $(0, 7)$ .
Axis of Symmetry: The axis of symmetry of the parabola is the vertical line $x = 0$ .
Direction: The parabola opens downwards, indicating that the vertex is the maximum point of the function.
Width: The width of the parabola is determined by the coefficient of the squared term, $a = -6$ .

Q: What is the significance of the vertex of the parabola? A: The vertex of the parabola is the maximum point of the function, indicating that the parabola opens downwards.

Q: How do I determine the direction of the parabola? A: You can determine the direction of the parabola by looking at the coefficient of the squared term, $a$ . If $a$ is positive, the parabola opens upwards. If $a$ is negative, the parabola opens downwards.

Q: What is the significance of the axis of symmetry of the parabola? A: The axis of symmetry of the parabola is the vertical line that passes through the vertex of the parabola. It is a line of symmetry that divides the parabola into two equal parts.

Q: How do I graph the parabola using a graphing calculator? A: To graph the parabola using a graphing calculator, you need to enter the function into the calculator and adjust the window settings to view the graph.

Conclusion

In this article, we answered several questions related to graphing the function $f(x) = -6x^2 + 7$ . We discussed the key features of the graph of the parabola, including the vertex, axis of symmetry, direction, and width. We also provided tips on how to graph the parabola using a graphing calculator.