Jason kept track of the number of songs he downloaded each week for 10 consecutive weeks. This function models the number of songs he downloaded, where $x$ is the number of weeks:
[tex]N(x)=-3(x-4)^2+60[/tex]
Plot the axis of symmetry and the vertex for this function.
Answer (2)
Identify the vertex form of the quadratic function: N ( x ) = − 3 ( x − 4 ) 2 + 60 .
Determine the vertex: The vertex is ( 4 , 60 ) .
Find the axis of symmetry: The axis of symmetry is x = 4 .
Plot the vertex and axis of symmetry on the coordinate plane: Vertex ( 4 , 60 ) , Axis of symmetry x = 4 .
Explanation
Identifying the Vertex Form The given function is N ( x ) = − 3 ( x − 4 ) 2 + 60 . This is a quadratic function in vertex form, which is f ( x ) = a ( x − h ) 2 + k , where ( h , k ) is the vertex of the parabola.
Finding the Vertex Comparing N ( x ) = − 3 ( x − 4 ) 2 + 60 with the vertex form f ( x ) = a ( x − h ) 2 + k , we can identify the vertex. Here, h = 4 and k = 60 . Therefore, the vertex of the parabola is ( 4 , 60 ) .
Determining the Axis of Symmetry The axis of symmetry for a parabola in vertex form is a vertical line given by x = h . Since the vertex is ( 4 , 60 ) , the axis of symmetry is the vertical line x = 4 .
Plotting the Vertex and Axis of Symmetry To plot the vertex and the axis of symmetry, first locate the point (4, 60) on the coordinate plane. This point is the vertex of the parabola. Then, draw a vertical line that passes through x = 4 . This line is the axis of symmetry.
Final Answer The vertex is ( 4 , 60 ) and the axis of symmetry is x = 4 .
Examples
Understanding quadratic functions like the one in this problem can help in various real-world scenarios. For example, if you're launching a projectile, the height of the projectile over time can be modeled by a quadratic function. The vertex of the parabola would represent the maximum height the projectile reaches, and the axis of symmetry would indicate the time at which the maximum height is achieved. Similarly, businesses can use quadratic functions to model profit, where the vertex represents the maximum profit and the axis of symmetry indicates the point at which resources are optimally allocated. Let's say the profit P ( x ) from selling x items is given by P ( x ) = − 0.5 ( x − 50 ) 2 + 1250 . The maximum profit is $1250 when selling 50 items.
The vertex of the function is (4, 60), which indicates the maximum downloads Jason achieved in week 4. The axis of symmetry is given by the line x = 4. This means that the parabola is symmetric around this line, and it helps identify the maximum point in the graph.
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