Question

which equation represents a graph with a vertex at (-3, 2)?
y = 4x² + 24x + 38
y = 4x² - 24x + 38
y = 4x² + 12x + 2
y = 4x² + 18x + 13

Explanation:

Step1: Recall vertex formula

For a quadratic \( y = ax^2 + bx + c \), vertex \( x \)-coordinate is \( x = -\frac{b}{2a} \). We check each option.

Step2: Check Option 1 (\( y = 4x^2 + 24x + 38 \))

Here, \( a = 4 \), \( b = 24 \).
\( x = -\frac{24}{2 \times 4} = -3 \).
Now find \( y \) at \( x = -3 \):
\( y = 4(-3)^2 + 24(-3) + 38 = 4(9) - 72 + 38 = 36 - 72 + 38 = 2 \).
Vertex is \((-3, 2)\), matches. Let's confirm others.

Step3: Check Option 2 (\( y = 4x^2 - 24x + 38 \))

\( x = -\frac{-24}{2 \times 4} = 3 \). Not \(-3\), eliminate.

Step4: Check Option 3 (\( y = 4x^2 + 12x + 2 \))

\( x = -\frac{12}{2 \times 4} = -\frac{3}{2} \). Not \(-3\), eliminate.

Step5: Check Option 4 (\( y = 4x^2 + 18x + 13 \))

\( x = -\frac{18}{2 \times 4} = -\frac{9}{4} \). Not \(-3\), eliminate.

Answer:

A. \( y = 4x^2 + 24x + 38 \) (assuming the first option is labeled A, as per typical multiple-choice; if labels differ, adjust, but based on calculation, the first equation is correct)