Which equation represents a graph with a vertex at $(1,-6)$?
A. $y=3 x^2+6 x-3$
B. $y=3 x^2-6 x-3$
C. $y=3 x^2-8 x-1$
D. $y=3 x^2-3 x-6$
Answer (2)
Find the vertex of each quadratic equation using the formula h = − 2 a b and k = f ( h ) .
Calculate the vertex for y = 3 x 2 + 6 x − 3 , which is ( − 1 , − 6 ) .
Calculate the vertex for y = 3 x 2 − 6 x − 3 , which is ( 1 , − 6 ) .
The equation with vertex ( 1 , − 6 ) is y = 3 x 2 − 6 x − 3 , so the answer is y = 3 x 2 − 6 x − 3 .
Explanation
Understanding the Problem We are given four quadratic equations and asked to identify the one with a vertex at ( 1 , − 6 ) . The vertex form of a quadratic equation is given by y = a ( x − h ) 2 + k , where ( h , k ) represents the vertex of the parabola. Our goal is to find the equation that, when converted to vertex form, has h = 1 and k = − 6 .
Finding the Vertex To find the vertex of each quadratic equation, we can use the formula h = − 2 a b and k = f ( h ) , where y = a x 2 + b x + c is the standard form of the quadratic equation. Alternatively, we can complete the square to convert each equation to vertex form y = a ( x − h ) 2 + k .
Calculating Vertices Let's analyze each equation:
y = 3 x 2 + 6 x − 3
h = − 2 ( 3 ) 6 = − 1
k = 3 ( − 1 ) 2 + 6 ( − 1 ) − 3 = 3 − 6 − 3 = − 6
Vertex: ( − 1 , − 6 )
y = 3 x 2 − 6 x − 3
h = − 2 ( 3 ) − 6 = 1
k = 3 ( 1 ) 2 − 6 ( 1 ) − 3 = 3 − 6 − 3 = − 6
Vertex: ( 1 , − 6 )
y = 3 x 2 − 8 x − 1
h = − 2 ( 3 ) − 8 = 3 4
k = 3 ( 3 4 ) 2 − 8 ( 3 4 ) − 1 = 3 ( 9 16 ) − 3 32 − 1 = 3 16 − 3 32 − 3 3 = − 3 19
Vertex: ( 3 4 , − 3 19 )
y = 3 x 2 − 3 x − 6
h = − 2 ( 3 ) − 3 = 2 1
k = 3 ( 2 1 ) 2 − 3 ( 2 1 ) − 6 = 3 ( 4 1 ) − 2 3 − 6 = 4 3 − 4 6 − 4 24 = − 4 27
Vertex: ( 2 1 , − 4 27 )
Identifying the Correct Equation Comparing the vertices of each equation with the given vertex ( 1 , − 6 ) , we find that the equation y = 3 x 2 − 6 x − 3 has the vertex ( 1 , − 6 ) .
Final Answer Therefore, the equation that represents a graph with a vertex at ( 1 , − 6 ) is y = 3 x 2 − 6 x − 3 .
Examples
Understanding quadratic functions and their vertices is crucial in various real-world applications. For instance, when designing a parabolic reflector for a flashlight or satellite dish, engineers need to determine the vertex to optimize focus and signal strength. Similarly, in projectile motion, the vertex represents the maximum height reached by an object, which is vital for calculating trajectories in sports or military applications. By mastering the vertex form, students can solve optimization problems, model physical phenomena, and make informed decisions in engineering and physics contexts. For example, the equation y = 3 x 2 − 6 x − 3 can model the trajectory of a ball, where the vertex (1, -6) indicates the ball's lowest point is at x=1 and y=-6.
The equation representing a graph with a vertex at (1, -6) is option B: y = 3 x 2 − 6 x − 3 . This was determined by calculating the vertex for each quadratic equation using the vertex formulas. Only option B yielded the vertex coordinates (1, -6).
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