Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
![## Parabolic Function Graph Explanation
### Graph Description
This graph represents a parabolic function, which is a type of quadratic function. The graph is plotted on a Cartesian coordinate system with the x-axis (horizontal) and y-axis (vertical).
### Key Points on the Graph
The graph includes several key points marked and labeled:
- The vertex of the parabola is at the origin, denoted as (0,0).
- The graph passes through points (−1,1) and (1,1), which provide symmetry about the y-axis.
- Additional points on the parabola include (−2,4) and (2,4), indicating the function's value at these x-coordinates.
### Parabola Properties
- **Symmetry:** This parabola is symmetric about the y-axis. This can be seen as for every point (x, y) on one side of the y-axis, there is a corresponding point (−x, y) on the opposite side.
- **Vertex:** The vertex at (0,0) is the lowest point on the graph, indicating that the parabola opens upwards.
- **Direction:** The parabolic arms open upwards, which suggests that the quadratic term \( ax^2 \) has a positive coefficient.
### Interactive Feature
At the bottom of the graph image, there is an interactive feature indicating a target submission:
- "0/5 targets" which likely signifies an interactive exercise where users need to identify or interact with certain points on the parabola.
- A "Submit" button is present, which can be used to finalize user input regarding the targets.
In summary, this graph is a visual representation of a quadratic function, specifically a simple parabola with its vertex at the origin and various symmetric points across the y-axis. The interactive aspect at the bottom invites learners to engage with the graph further.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc3fa423d-5b51-4fe0-bc38-d9eeab964134%2Fcb286ead-9364-4c75-bc3a-b231d3f20043%2Fdc177un_processed.jpeg&w=3840&q=75)
![**The graph of \( f(x) = x^2 \) is shown below with five specific points labeled. Plot the location of each of these five points after the graph is reflected vertically.**
In this task, you are presented with a quadratic function \( f(x) = x^2 \). The graph of this function is a parabola that opens upwards. The current graph has five points that are labeled for reference.
After reflecting this graph vertically, each y-coordinate of the labeled points will be transformed. If a point on the original graph is at coordinates (x, y), reflecting it vertically will result in a new point at (x, -y).
**Steps to complete the task:**
1. Identify the coordinates of the five labeled points on the original graph.
2. Reflect each point vertically by changing the sign of the y-coordinate.
3. Plot the new set of points on the graph accordingly.
**Example:**
If one of the labeled points on the original graph is (2, 4):
- The y-coordinate is 4.
- After vertical reflection, the new y-coordinate will be -4.
- So, the new point will be (2, -4).
Apply this process to all five labeled points to complete the reflection.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc3fa423d-5b51-4fe0-bc38-d9eeab964134%2Fcb286ead-9364-4c75-bc3a-b231d3f20043%2Fl9yoird_processed.jpeg&w=3840&q=75)
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