Which graph represents the equation $y=-(x-1)^2+1$?
Answer (2)
The equation is a parabola in vertex form y = a ( x − h ) 2 + k with vertex at ( h , k ) .
Identify the vertex as ( 1 , 1 ) and note that the parabola opens downwards since a = − 1 .
Calculate the x-intercepts by setting y = 0 , which gives x = 0 and x = 2 .
Find the y-intercept by setting x = 0 , resulting in y = 0 .
The graph is a downward-opening parabola with vertex ( 1 , 1 ) , x-intercepts at 0 and 2 , and y-intercept at 0 .
Explanation
Analyze the equation We are given the equation y = − ( x − 1 ) 2 + 1 and we need to determine which graph represents this equation. Let's analyze the properties of the equation to identify the correct graph.
Identify the vertex The given equation is in vertex form y = a ( x − h ) 2 + k , where ( h , k ) is the vertex of the parabola. In our case, a = − 1 , h = 1 , and k = 1 . Therefore, the vertex of the parabola is ( 1 , 1 ) .
Determine the direction Since the coefficient a = − 1 is negative, the parabola opens downwards.
Find the x-intercepts To find the x-intercepts, we set y = 0 and solve for x :
0 = − ( x − 1 ) 2 + 1 ( x − 1 ) 2 = 1 Taking the square root of both sides: x − 1 = ± 1 So, x = 1 + 1 = 2 or x = 1 − 1 = 0 . The x-intercepts are x = 0 and x = 2 .
Find the y-intercept To find the y-intercept, we set x = 0 and solve for y :
y = − ( 0 − 1 ) 2 + 1 y = − ( − 1 ) 2 + 1 y = − 1 + 1 y = 0 The y-intercept is y = 0 .
Conclusion Based on our analysis, the parabola has a vertex at ( 1 , 1 ) , opens downwards, has x-intercepts at x = 0 and x = 2 , and a y-intercept at y = 0 . Therefore, the graph representing the equation y = − ( x − 1 ) 2 + 1 is a parabola opening downwards with vertex (1,1) and x-intercepts 0 and 2.
Examples
Understanding parabolas is crucial in various real-world applications. For instance, the trajectory of a projectile, like a ball thrown in the air, follows a parabolic path. The equation y = − ( x − 1 ) 2 + 1 can model such a trajectory, where 'y' represents the height of the ball and 'x' represents the horizontal distance. The vertex (1,1) indicates the maximum height the ball reaches, and the x-intercepts (0 and 2) show where the ball starts and lands. This understanding helps in predicting the range and maximum height of projectiles in sports and engineering.
The graph for the equation y = − ( x − 1 ) 2 + 1 is a downward-opening parabola with its vertex at (1, 1) and x-intercepts at 0 and 2. Additionally, it has a y-intercept at 0. Thus, look for a graph that matches these characteristics.
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