Choose the graph of $y=(x-3)^2+1$.
Answer (1)
The equation y = ( x − 3 ) 2 + 1 represents a parabola. The vertex of the parabola is ( 3 , 1 ) , and since the coefficient of the x 2 term is positive, the parabola opens upwards. Therefore, the graph is a parabola with vertex at ( 3 , 1 ) opening upwards.
Explanation
Analyze the equation The given equation is y = ( x − 3 ) 2 + 1 . This is a quadratic equation in vertex form, which is y = a ( x − h ) 2 + k , where ( h , k ) is the vertex of the parabola. In this case, we have a = 1 , h = 3 , and k = 1 . Therefore, the vertex of the parabola is ( 3 , 1 ) . Since 0"> a = 1 > 0 , the parabola opens upwards.
Identify the vertex and direction The vertex of the parabola is ( 3 , 1 ) , and it opens upwards. We need to choose the graph that satisfies these conditions.
Conclusion The graph of the equation y = ( x − 3 ) 2 + 1 is a parabola with vertex at ( 3 , 1 ) that opens upwards.
Examples
Understanding parabolas is crucial in various fields, such as physics and engineering. For example, the trajectory of a projectile (like a ball thrown in the air) follows a parabolic path. The equation y = ( x − 3 ) 2 + 1 can represent such a trajectory, where y is the height of the projectile and x is the horizontal distance. The vertex (3,1) would then represent the highest point the projectile reaches. By understanding the properties of parabolas, we can predict the motion of objects and design structures that can withstand certain forces.
