Which equation represents a graph with a vertex at $(-3,2)$?

$y=4 x^2+24 x+38$

$y=4 x^2-24 x+38$

$y=4 x^2+12 x+2$

$y=4 x^2+16 x+13$

Convert each quadratic equation to vertex form y = a ( x − h ) 2 + k .
The first equation y = 4 x 2 + 24 x + 38 converts to y = 4 ( x + 3 ) 2 + 2 , with vertex ( − 3 , 2 ) .
The other equations do not have the vertex ( − 3 , 2 ) .
Therefore, the equation representing a graph with a vertex at ( − 3 , 2 ) is y = 4 x 2 + 24 x + 38 ​ .

Explanation

Understanding the Problem We are given four quadratic equations and asked to find the one whose graph has 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. We need to convert each given equation into vertex form and check which one has h = − 3 and k = 2 .

Analyzing 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 .

Simplify: y = 4 ( x + 3 ) 2 − 36 + 38 = 4 ( x + 3 ) 2 + 2 .


Thus, the vertex form of the first equation is y = 4 ( x + 3 ) 2 + 2 . The vertex is ( − 3 , 2 ) .

Analyzing the Second Equation Now let's analyze the second equation: y = 4 x 2 − 24 x + 38 . Completing the square:

Factor out the coefficient of x 2 : y = 4 ( x 2 − 6 x ) + 38 .

Complete the square: x 2 − 6 x + ( − 6/2 ) 2 = x 2 − 6 x + 9 = ( x − 3 ) 2 .

Add and subtract the necessary term: y = 4 ( x 2 − 6 x + 9 − 9 ) + 38 = 4 (( x − 3 ) 2 − 9 ) + 38 .

Simplify: y = 4 ( x − 3 ) 2 − 36 + 38 = 4 ( x − 3 ) 2 + 2 .


Thus, the vertex form of the second equation is y = 4 ( x − 3 ) 2 + 2 . The vertex is ( 3 , 2 ) .

Analyzing the Third Equation Now let's analyze the third equation: y = 4 x 2 + 12 x + 2 . Completing the square:

Factor out the coefficient of x 2 : y = 4 ( x 2 + 3 x ) + 2 .

Complete the square: x 2 + 3 x + ( 3/2 ) 2 = x 2 + 3 x + 9/4 = ( x + 3/2 ) 2 .

Add and subtract the necessary term: y = 4 ( x 2 + 3 x + 9/4 − 9/4 ) + 2 = 4 (( x + 3/2 ) 2 − 9/4 ) + 2 .

Simplify: y = 4 ( x + 3/2 ) 2 − 9 + 2 = 4 ( x + 3/2 ) 2 − 7 .


Thus, the vertex form of the third equation is y = 4 ( x + 3/2 ) 2 − 7 . The vertex is ( − 3/2 , − 7 ) .

Analyzing the Fourth Equation Now let's analyze the fourth equation: y = 4 x 2 + 16 x + 13 . Completing the square:

Factor out the coefficient of x 2 : y = 4 ( x 2 + 4 x ) + 13 .

Complete the square: x 2 + 4 x + ( 4/2 ) 2 = x 2 + 4 x + 4 = ( x + 2 ) 2 .

Add and subtract the necessary term: y = 4 ( x 2 + 4 x + 4 − 4 ) + 13 = 4 (( x + 2 ) 2 − 4 ) + 13 .

Simplify: y = 4 ( x + 2 ) 2 − 16 + 13 = 4 ( x + 2 ) 2 − 3 .


Thus, the vertex form of the fourth equation is y = 4 ( x + 2 ) 2 − 3 . The vertex is ( − 2 , − 3 ) .

Conclusion Comparing the vertices of the four equations with the given vertex ( − 3 , 2 ) , we find that the first equation, y = 4 x 2 + 24 x + 38 , has the vertex ( − 3 , 2 ) . Therefore, it is the correct equation.

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 for satellite dishes or solar cookers, ensuring the focus is at the desired point. Similarly, in projectile motion, the vertex represents the maximum height reached by an object, which is vital in sports like basketball or launching projectiles in physics experiments. By mastering the vertex form, one can easily determine key characteristics of parabolic trajectories and optimize designs accordingly.

Answered by GinnyAnswer | 2025-07-04