Select the correct answer.
The table represents quadratic function [tex]$g$[/tex]. Which statement is true about the function?
| x | -5 | -4 | -3 | -2 | -1 | 0 |
|---|----|----|----|----|----|---|
| g(x) | -1 | 0 | -1 | -4 | -9 | -16 |
A. The maximum occurs at the function's [tex]$y$[/tex]-intercept.
B. The minimum occurs at the function's [tex]$y$[/tex]-intercept.
C. The maximum occurs at the function's [tex]$x$[/tex]-intercept.
D. The minimum occurs at the function's [tex]$x$[/tex]-intercept.
Answer (2)
The vertex of the quadratic function is identified as ( − 4 , 0 ) from the table.
The vertex is determined to be a maximum since the function values decrease as x moves away from − 4 .
The x -intercept is − 4 and the y -intercept is − 16 .
The maximum occurs at the x -intercept, so the correct answer is C .
Explanation
Problem Analysis We are given a table of values for a quadratic function g ( x ) and asked to determine a true statement about the function's maximum or minimum and its location relative to the x and y intercepts.
Identifying the Vertex From the table, we have the following points: ( − 5 , − 1 ) , ( − 4 , 0 ) , ( − 3 , − 1 ) , ( − 2 , − 4 ) , ( − 1 , − 9 ) , ( 0 , − 16 ) .
We can observe that the function values decrease as x moves away from x = − 4 . This suggests that the vertex of the parabola is at x = − 4 .
Determining Maximum or Minimum Since g ( − 4 ) = 0 , the vertex is at the point ( − 4 , 0 ) .
To determine if the vertex is a maximum or minimum, we can look at the values around x = − 4 . Since g ( − 5 ) = − 1 and g ( − 3 ) = − 1 , and these values are less than g ( − 4 ) = 0 , the vertex is a maximum.
Finding Intercepts The x -intercept is the point where g ( x ) = 0 . From the table, we see that g ( − 4 ) = 0 , so the x -intercept is − 4 .
The y -intercept is the point where x = 0 . From the table, we see that g ( 0 ) = − 16 , so the y -intercept is − 16 .
Checking Options The maximum of the function occurs at the vertex, which is at ( − 4 , 0 ) . This is also the x -intercept. Now we check the given options: A. The maximum occurs at the function's y -intercept. (False, the maximum occurs at the x -intercept.) B. The minimum occurs at the function's y -intercept. (False, the maximum occurs at the x -intercept.) C. The maximum occurs at the function's x -intercept. (True) D. The minimum occurs at the function's x -intercept. (False, the maximum occurs at the x -intercept.)
Conclusion Therefore, the correct statement is that the maximum occurs at the function's x -intercept.
Examples
Understanding quadratic functions is crucial in various real-world applications. For instance, engineers use quadratic equations to model the trajectory of projectiles, such as rockets or balls. By determining the maximum height (vertex) of the trajectory, they can optimize launch angles and predict landing points. Similarly, in business, quadratic functions can model profit curves, where the maximum profit point helps companies determine optimal production levels.
The maximum of the quadratic function occurs at the x -intercept, which is at ( − 4 , 0 ) . The chosen correct answer is C. Due to the function's behavior, there is no minimum since the vertex represents a maximum value.
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