The graph of a quadratic function passed through the point $(3.5,0)$ and has a turning point at $(2.75,-1)$. Find the coordinate of the other root.
Answer (2)
Express the quadratic function in vertex form using the turning point: f ( x ) = a ( x − 2.75 ) 2 − 1 .
Use the given point ( 3.5 , 0 ) to find the leading coefficient a = 9 16 .
Solve for the roots by setting f ( x ) = 0 and finding x = 2.75 ± 0.75 .
The other root is x = 2 .
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Explanation
Understanding the Problem We are given that a quadratic function passes through the point ( 3.5 , 0 ) and has a turning point at ( 2.75 , − 1 ) . Our goal is to find the coordinate of the other root of this quadratic function.
Using Vertex Form Since we know the turning point (vertex) of the quadratic function, we can express the function in vertex form: f ( x ) = a ( x − h ) 2 + k , where ( h , k ) is the vertex. In our case, the vertex is ( 2.75 , − 1 ) , so we have f ( x ) = a ( x − 2.75 ) 2 − 1 .
Finding the Leading Coefficient We are also given that the function passes through the point ( 3.5 , 0 ) . We can use this information to find the value of a . Substituting x = 3.5 and f ( x ) = 0 into the equation, we get: 0 = a ( 3.5 − 2.75 ) 2 − 1 .
Calculating a Now, let's solve for a : 0 = a ( 0.75 ) 2 − 1
1 = a ( 0.5625 )
a = 0.5625 1 = 16 9 1 = 9 16 . So, our quadratic function is f ( x ) = 9 16 ( x − 2.75 ) 2 − 1 .
Solving for the Roots To find the roots of the quadratic function, we set f ( x ) = 0 and solve for x : 9 16 ( x − 2.75 ) 2 − 1 = 0
9 16 ( x − 2.75 ) 2 = 1
( x − 2.75 ) 2 = 16 9
Taking the square root of both sides, we get: x − 2.75 = ± 4 3 = ± 0.75
So, x = 2.75 ± 0.75 .
Finding the Other Root We have two possible values for x : x 1 = 2.75 + 0.75 = 3.5 (which we already knew) x 2 = 2.75 − 0.75 = 2 . Therefore, the other root is x = 2 .
Using Symmetry Alternatively, we can use the symmetry property of parabolas. The axis of symmetry is at x = 2.75 . The given root is at x = 3.5 , which is 3.5 − 2.75 = 0.75 units away from the axis of symmetry. The other root will be 0.75 units away from the axis of symmetry on the other side. Thus, the other root is 2.75 − 0.75 = 2 .
Final Answer The coordinate of the other root is 2.
Examples
Quadratic functions are used in various real-world applications, such as modeling the trajectory of a projectile, designing parabolic mirrors and reflectors, and determining the optimal dimensions for maximizing area or minimizing cost. For example, if you're launching a rocket, understanding the roots (where the projectile hits the ground) and the vertex (the maximum height) is crucial for predicting its path and ensuring a successful landing. Similarly, in business, quadratic functions can help optimize pricing strategies to maximize profit, where the roots represent break-even points and the vertex represents the optimal price point.
The other root of the quadratic function is 2, found by using the vertex form and applying the known root and vertex information. This results in the roots being 2.75 ± 0.75. Thus, the final answer is 2 .
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