Use a graphing calculator to approximate the vertex of the graph of the parabola defined by the following equation:

[tex]$y=-5 x^2-4 x+6$[/tex]

A. (2,6.8)
B. (-0.4,6.8)
C. (0.4,6.8)
D. (-0.4,5.2)

Please select the best answer from the choices provided.

The vertex of the parabola given by the equation y = − 5 x 2 − 4 x + 6 is ( − 0.4 , 6.8 ) . This is found by calculating the x-coordinate using the formula x = − 2 a b ​ and then substituting back to find the y-coordinate. Therefore, the chosen answer is B: (-0.4, 6.8).
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Answered by Anonymous | 2025-07-04

Find the x-coordinate of the vertex using the formula x = − 2 a b ​ , where a = − 5 and b = − 4 , which gives x = − 0.4 .
Substitute x = − 0.4 into the equation y = − 5 x 2 − 4 x + 6 to find the y-coordinate.
Calculate y = − 5 ( − 0.4 ) 2 − 4 ( − 0.4 ) + 6 = 6.8 .
The vertex of the parabola is ( − 0.4 , 6.8 ) , so the answer is B ​ .

Explanation

Understanding the Problem We are given the equation of a parabola y = − 5 x 2 − 4 x + 6 and asked to find the vertex. The vertex of a parabola in the form y = a x 2 + b x + c can be found using the formula x = − 2 a b ​ for the x-coordinate, and then substituting this value back into the equation to find the y-coordinate.

Finding the x-coordinate of the Vertex In our equation, a = − 5 and b = − 4 . Plugging these values into the vertex formula, we get: x = − 2 ( − 5 ) − 4 ​ = − 10 4 ​ = − 0.4

Finding the y-coordinate of the Vertex Now we substitute x = − 0.4 back into the equation to find the y-coordinate: y = − 5 ( − 0.4 ) 2 − 4 ( − 0.4 ) + 6
y = − 5 ( 0.16 ) + 1.6 + 6
y = − 0.8 + 1.6 + 6
y = 0.8 + 6
y = 6.8

Final Answer Therefore, the vertex of the parabola is ( − 0.4 , 6.8 ) . Comparing this to the given options, we see that option b, ( − 0.4 , 6.8 ) , is the correct answer.


Examples
Understanding parabolas and their vertices is crucial in many real-world applications. For example, engineers use this knowledge to design suspension bridges, where the cables often form parabolic shapes. The vertex helps determine the lowest point of the cable, which is essential for calculating stress and ensuring the bridge's stability. Similarly, in sports, understanding the parabolic trajectory of a ball helps athletes optimize their throws or kicks for maximum distance or accuracy. Even in economics, parabolas can model cost curves, where the vertex represents the point of minimum cost.

Answered by GinnyAnswer | 2025-07-04