Which equation represents a graph with a vertex at $(-3,2)$?
$y=4 x^2+24 x+38$
$y=4 x^2-24 x+38$
$y=4 x^2+12 x+2$
$y=4 x^2+16 x+13$
Answer (2)
Convert the given quadratic equation to vertex form y = a ( x − h ) 2 + k .
Complete the square for the equation y = 4 x 2 + 24 x + 38 , resulting in y = 4 ( x + 3 ) 2 + 2 .
Identify the vertex as ( − 3 , 2 ) by comparing the equation to the vertex form.
Conclude that the equation representing a graph with a vertex at ( − 3 , 2 ) is y = 4 x 2 + 24 x + 38 .
Explanation
Understanding the Problem and Strategy We are given four quadratic equations and asked to identify the one with a vertex at ( − 3 , 2 ) . The vertex form of a quadratic equation is given by y = a ( x − h ) 2 + k , where ( h , k ) is the vertex of the parabola. Our goal is to rewrite each given equation in vertex form and check which one has h = − 3 and k = 2 .
Completing the Square for the First Equation Let's analyze the first equation: y = 4 x 2 + 24 x + 38 . To convert it to vertex form, we complete the square:
Factor out the coefficient of x 2 from the first two terms: y = 4 ( x 2 + 6 x ) + 38 .
Complete the square inside the parenthesis: x 2 + 6 x + ( 6/2 ) 2 = x 2 + 6 x + 9 = ( x + 3 ) 2 .
Add and subtract the necessary term to complete the square: y = 4 ( x 2 + 6 x + 9 − 9 ) + 38 = 4 (( x + 3 ) 2 − 9 ) + 38 .
Distribute and simplify: y = 4 ( x + 3 ) 2 − 36 + 38 = 4 ( x + 3 ) 2 + 2 .
Identifying the Vertex The vertex form of the first equation is y = 4 ( x + 3 ) 2 + 2 . Comparing this to y = a ( x − h ) 2 + k , we see that h = − 3 and k = 2 . Therefore, the vertex is ( − 3 , 2 ) .
Final Answer Since the first equation, y = 4 x 2 + 24 x + 38 , has a vertex at ( − 3 , 2 ) , it is the correct answer. We can verify the other options to be sure, but it's not strictly necessary.
Verification of Other Options For completeness, let's quickly check the other equations:
y = 4 x 2 − 24 x + 38 = 4 ( x 2 − 6 x ) + 38 = 4 ( x 2 − 6 x + 9 ) + 38 − 36 = 4 ( x − 3 ) 2 + 2 . Vertex is ( 3 , 2 ) .
y = 4 x 2 + 12 x + 2 = 4 ( x 2 + 3 x ) + 2 = 4 ( x 2 + 3 x + 9/4 ) + 2 − 9 = 4 ( x + 3/2 ) 2 − 7 . Vertex is ( − 3/2 , − 7 ) .
y = 4 x 2 + 16 x + 13 = 4 ( x 2 + 4 x ) + 13 = 4 ( x 2 + 4 x + 4 ) + 13 − 16 = 4 ( x + 2 ) 2 − 3 . Vertex is ( − 2 , − 3 ) .
Examples
Understanding quadratic equations and their vertex form is crucial in various real-world applications. For instance, engineers use this knowledge to design parabolic reflectors in satellite dishes or to model the trajectory of projectiles. By knowing the vertex, they can determine the maximum or minimum value of a function, which is essential for optimizing designs and predicting outcomes. For example, if you want to throw a ball as far as possible, understanding the parabolic trajectory and its vertex helps you determine the optimal angle and initial velocity to achieve maximum range. The equation y = 4 x 2 + 24 x + 38 can model a physical phenomenon, and finding its vertex helps identify key characteristics of that phenomenon.
The equation that represents a graph with a vertex at ( − 3 , 2 ) is y = 4 x 2 + 24 x + 38 . This was determined by converting each provided quadratic equation into vertex form. The first equation matched the vertex criteria perfectly.
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