What are the coordinates of the vertex of the parabola $y=x^2-2 x-1$?
A. $(-1,2)$
B. $(1,2)$
C. $(-1,-2)$
D. $(1,-2)
Answer (2)
The vertex of the parabola given by y = x 2 − 2 x − 1 is at the coordinates ( 1 , − 2 ) . Therefore, the correct answer is option D. This is obtained by calculating the vertex using the formula for the x-coordinate and substituting back to find the y-coordinate.
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Identify the coefficients of the quadratic equation: a = 1 , b = − 2 , c = − 1 .
Calculate the x-coordinate of the vertex using the formula: x = − 2 a b = 1 .
Substitute the x-coordinate into the equation to find the y-coordinate: y = ( 1 ) 2 − 2 ( 1 ) − 1 = − 2 .
The coordinates of the vertex are ( 1 , − 2 ) .
Explanation
Understanding the Problem We are given the equation of a parabola y = x 2 − 2 x − 1 and asked to find the coordinates of its 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.
Identifying Coefficients In our equation, y = x 2 − 2 x − 1 , we can identify the coefficients as a = 1 , b = − 2 , and c = − 1 .
Calculating the x-coordinate Now, we can find the x-coordinate of the vertex using the formula x = − 2 a b . Substituting the values of a and b , we get: x = − 2 ( 1 ) − 2 = 2 2 = 1
Calculating the y-coordinate Next, we substitute the x-coordinate, x = 1 , back into the equation y = x 2 − 2 x − 1 to find the y-coordinate: y = ( 1 ) 2 − 2 ( 1 ) − 1 = 1 − 2 − 1 = − 2
Finding the Vertex Therefore, the coordinates of the vertex of the parabola are ( 1 , − 2 ) .
Examples
Understanding the vertex of a parabola is useful in many real-world applications. For example, if you're launching a projectile, the vertex represents the maximum height the projectile will reach. Similarly, in business, if you have a cost function that is a parabola, the vertex can represent the point at which you minimize your costs. Knowing how to find the vertex allows you to optimize various processes and make informed decisions.
