Factoring out the coefficient of x

Oct 13, 2025 by magicalceylontours.com 102 views

How is the Graph of the Parent Quadratic Function Transformed to Produce the Graph of $y=-(2x+6)^2+3$ ?

Understanding the Parent Quadratic Function

The parent quadratic function is a quadratic function in the form of $y = ax^2 + bx + c$ . In this case, the parent quadratic function is $y = -(x+3)^2 + 3$ . The graph of this function is a parabola that opens downwards, with its vertex at the point $(-3, 3)$ .

Transforming the Parent Quadratic Function

To produce the graph of $y=-(2x+6)^2+3$ , we need to apply a series of transformations to the parent quadratic function. These transformations include:

Horizontal Compression: The graph is compressed horizontally by a factor of 2. This means that the x-coordinates of the graph are halved.
Shift Left: The graph is shifted left by 3 units. This means that the x-coordinates of the graph are decreased by 3.
Reflection over the x-axis: The graph is reflected over the x-axis. This means that the y-coordinates of the graph are negated.
Translation Up: The graph is translated up by 3 units. This means that the y-coordinates of the graph are increased by 3.

Applying the Transformations

To apply these transformations, we need to rewrite the parent quadratic function in the form of $y = -(2x+6)^2 + 3$ . We can do this by:

Factoring out the coefficient of x: The coefficient of x in the parent quadratic function is 1. We can factor this out to get $y = -(x+3)^2 + 3$ .
Applying the horizontal compression: We can apply the horizontal compression by multiplying the x-coordinates by 2. This gives us $y = -(2x+6)^2 + 3$ .
Applying the shift left: We can apply the shift left by decreasing the x-coordinates by 3. This gives us $y = -(2(x-3)+6)^2 + 3$ .
Applying the reflection over the x-axis: We can apply the reflection over the x-axis by negating the y-coordinates. This gives us $y = -(2(x-3)+6)^2 - 3$ .
Applying the translation up: We can apply the translation up by increasing the y-coordinates by 3. This gives us $y = -(2(x-3)+6)^2 + 3$ .

Conclusion

In conclusion, the graph of the parent quadratic function is transformed to produce the graph of $y=-(2x+6)^2+3$ by applying a series of transformations. These transformations include horizontal compression, shift left, reflection over the x-axis, and translation up. By applying these transformations, we can rewrite the parent quadratic function in the form of $y = -(2x+6)^2 + 3$ .

Understanding the Graph of $y=-(2x+6)^2+3$

The graph of $y=-(2x+6)^2+3$ is a parabola that opens downwards. The vertex of the graph is at the point $(-3, 3)$ . The graph is compressed horizontally by a factor of 2, shifted left by 3 units, reflected over the x-axis, and translated up by 3 units.

Key Features of the Graph

Vertex: The vertex of the graph is at the point $(-3, 3)$ .
Axis of Symmetry: The axis of symmetry of the graph is the vertical line $x = -3$ .
Direction of Opening: The graph opens downwards.
X-intercepts: The x-intercepts of the graph are at the points $(-6, 0)$ and $(-0, 0)$ .
Y-intercept: The y-intercept of the graph is at the point $(0, 3)$ .

Graphing the Function

To graph the function $y=-(2x+6)^2+3$ , we can use the following steps:

Plot the vertex: Plot the vertex of the graph at the point $(-3, 3)$ .
Plot the axis of symmetry: Plot the axis of symmetry of the graph as the vertical line $x = -3$ .
Plot the x-intercepts: Plot the x-intercepts of the graph at the points $(-6, 0)$ and $(-0, 0)$ .
Plot the y-intercept: Plot the y-intercept of the graph at the point $(0, 3)$ .
Plot additional points: Plot additional points on the graph to complete it.

Conclusion

In conclusion, the graph of $y=-(2x+6)^2+3$ is a parabola that opens downwards. The vertex of the graph is at the point $(-3, 3)$ . The graph is compressed horizontally by a factor of 2, shifted left by 3 units, reflected over the x-axis, and translated up by 3 units. By understanding the key features of the graph and using the steps to graph the function, we can visualize the graph of $y=-(2x+6)^2+3$ .
Q&A: Understanding the Graph of $y=-(2x+6)^2+3$

Q: What is the parent quadratic function?

A: The parent quadratic function is a quadratic function in the form of $y = ax^2 + bx + c$ . In this case, the parent quadratic function is $y = -(x+3)^2 + 3$ .

Q: What are the transformations applied to the parent quadratic function to produce the graph of $y=-(2x+6)^2+3$ ?

A: The transformations applied to the parent quadratic function are:

Horizontal Compression: The graph is compressed horizontally by a factor of 2.
Shift Left: The graph is shifted left by 3 units.
Reflection over the x-axis: The graph is reflected over the x-axis.
Translation Up: The graph is translated up by 3 units.

Q: How do the transformations affect the graph of the parent quadratic function?

A: The transformations affect the graph of the parent quadratic function in the following ways:

Horizontal Compression: The graph is compressed horizontally by a factor of 2, which means that the x-coordinates of the graph are halved.
Shift Left: The graph is shifted left by 3 units, which means that the x-coordinates of the graph are decreased by 3.
Reflection over the x-axis: The graph is reflected over the x-axis, which means that the y-coordinates of the graph are negated.
Translation Up: The graph is translated up by 3 units, which means that the y-coordinates of the graph are increased by 3.

Q: What is the vertex of the graph of $y=-(2x+6)^2+3$ ?

A: The vertex of the graph of $y=-(2x+6)^2+3$ is at the point $(-3, 3)$ .

Q: What is the axis of symmetry of the graph of $y=-(2x+6)^2+3$ ?

A: The axis of symmetry of the graph of $y=-(2x+6)^2+3$ is the vertical line $x = -3$ .

Q: How do you graph the function $y=-(2x+6)^2+3$ ?

A: To graph the function $y=-(2x+6)^2+3$ , you can use the following steps:

Plot the vertex: Plot the vertex of the graph at the point $(-3, 3)$ .
Plot the axis of symmetry: Plot the axis of symmetry of the graph as the vertical line $x = -3$ .
Plot the x-intercepts: Plot the x-intercepts of the graph at the points $(-6, 0)$ and $(-0, 0)$ .
Plot the y-intercept: Plot the y-intercept of the graph at the point $(0, 3)$ .
Plot additional points: Plot additional points on the graph to complete it.

Q: What are the key features of the graph of $y=-(2x+6)^2+3$ ?

A: The key features of the graph of $y=-(2x+6)^2+3$ are:

Vertex: The vertex of the graph is at the point $(-3, 3)$ .
Axis of Symmetry: The axis of symmetry of the graph is the vertical line $x = -3$ .
Direction of Opening: The graph opens downwards.
X-intercepts: The x-intercepts of the graph are at the points $(-6, 0)$ and $(-0, 0)$ .
Y-intercept: The y-intercept of the graph is at the point $(0, 3)$ .

Q: How do you determine the x-intercepts of the graph of $y=-(2x+6)^2+3$ ?

A: To determine the x-intercepts of the graph of $y=-(2x+6)^2+3$ , you can set the function equal to 0 and solve for x.

Q: How do you determine the y-intercept of the graph of $y=-(2x+6)^2+3$ ?

A: To determine the y-intercept of the graph of $y=-(2x+6)^2+3$ , you can substitute x = 0 into the function and solve for y.

Q: What is the significance of the axis of symmetry in the graph of $y=-(2x+6)^2+3$ ?

A: The axis of symmetry in the graph of $y=-(2x+6)^2+3$ is the vertical line $x = -3$ . This line divides the graph into two equal halves, with the vertex of the graph located at the point of intersection.

Q: How do you use the axis of symmetry to graph the function $y=-(2x+6)^2+3$ ?

A: To use the axis of symmetry to graph the function $y=-(2x+6)^2+3$ , you can plot the axis of symmetry as the vertical line $x = -3$ and then plot the vertex of the graph at the point of intersection. From there, you can plot additional points on the graph to complete it.