To provide a detailed response, I'll need to assume some typical characteristics of a parabolic function based on its graph. Without an actual graph, I will explain how to identify key features of a function, and then give an example of a real-world application.
Part A: Key Features of the Graph
Assuming the graph is a parabola, typically:
A. Vertex: The vertex is the highest or lowest point of the parabola (depending on whether it opens upwards or downwards).
- Let's say the vertex is at (h, k). You would need to look at the graph to identify these coordinates accurately.
B. Axis of symmetry: This is a vertical line that passes through the vertex. The equation for the axis of symmetry would be x = h, where h is the x-coordinate of the vertex.
C. y-intercept: The y-intercept is the point where the graph crosses the y-axis. This occurs when x = 0. You would substitute x = 0 in the function f(x) to find f(0).
- Let’s assume y-intercept is (0, b).
D. x-intercepts: The x-intercepts are the points where the graph crosses the x-axis. These occur when f(x) = 0. If there are two intercepts, denote them as (x1, 0) and (x2, 0).
- For our example, it could be (x1, 0) and (x2, 0).
Part B: Real-World Example
Example Scenario: Projectile Motion
A real-world example that could be represented by a quadratic function is the motion of a projectile, such as a ball thrown into the air. The height of the ball at any time can be modeled using a quadratic function of the form:
\[ f(t) = -at^2 + bt + c \]
Where:
- \( t \) represents time,
- \( a \), \( b \), and \( c \) are constants that depend on the initial velocity and height of the throw, and the effects of gravity.
Key Features Representation:
-
Vertex: The vertex of the parabola represents the maximum height reached by the ball. The coordinates (h, k) would give the time at which this maximum height occurs and the maximum height itself.
-
Axis of symmetry: The axis of symmetry (x = h) shows the time at which the ball reaches its maximum height. The motion of the projectile is symmetric about this axis.
-
y-intercept: The y-intercept (0, b) represents the initial height of the ball when it is thrown (at time t = 0).
-
x-intercepts: The x-intercepts (x1, 0) and (x2, 0) represent the points in time when the ball hits the ground (when the height is zero).
By interpreting these features, one can analyze the behavior of the projectile throughout its flight, including how high it reaches and when it will land back on the ground.
Conclusion
In order to complete this task accurately, you would need to refer to the specific graph in question to pinpoint the exact coordinates and values mentioned.