Use a graphing calculator to approximate the vertex of the graph of the parabola defined by the following equation:
[tex]$y=x^2+2 x-9$[/tex]
A. $(-1,-10)$
B. $(-1,-9)$
C. $(1,10)$
D. $(1,-10)$
Answer (2)
The vertex of the parabola defined by the equation y = x 2 + 2 x − 9 is found to be ( − 1 , − 10 ) . Using the formula x = − 2 a b , the x-coordinate is determined first, and then by substituting this value into the equation, the y-coordinate is calculated. Thus, the selected answer is option A: ( − 1 , − 10 ) .
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Find the x-coordinate of the vertex using the formula x = − 2 a b , which gives x = − 1 .
Substitute x = − 1 into the equation y = x 2 + 2 x − 9 to find the y-coordinate.
Calculate y = ( − 1 ) 2 + 2 ( − 1 ) − 9 = 1 − 2 − 9 = − 10 .
The vertex of the parabola is therefore ( − 1 , − 10 ) .
Explanation
Understanding the Problem We are given the equation of a parabola y = x 2 + 2 x − 9 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 In our equation, y = x 2 + 2 x − 9 , we have a = 1 , b = 2 , and c = − 9 . Let's find the x-coordinate of the vertex: x = − 2 a b = − 2 ( 1 ) 2 = − 2 2 = − 1 So, the x-coordinate of the vertex is -1.
Finding the y-coordinate Now, we substitute x = − 1 into the equation to find the y-coordinate of the vertex: y = ( − 1 ) 2 + 2 ( − 1 ) − 9 = 1 − 2 − 9 = − 10 So, the y-coordinate of the vertex is -10.
Determining the Vertex Therefore, the vertex of the parabola is ( − 1 , − 10 ) . Comparing this to the given options, we see that option a. ( − 1 , − 10 ) is the correct answer.
Examples
Understanding parabolas is crucial in various real-world applications. For instance, engineers use parabolic shapes in designing suspension bridges and antennas to optimize weight distribution and signal reception. Architects also employ parabolic arches for their structural strength and aesthetic appeal. Moreover, in physics, projectile motion follows a parabolic path, which is essential for calculating trajectories in sports and ballistics. By mastering the properties of parabolas, students gain valuable insights into these practical applications and develop a deeper appreciation for the interconnectedness of mathematics and the world around them.
