Question

find the graph of the following and state whether there is a maximum or minimum. $y = x^2 + 5x + 6$ select the correct response: the graph has a maximum graph 1 the graph has a maximum. graph 2 the graph has a minimum. graph 3

Explanation:

Step1: Identify the parabola's direction

The given equation is \( y = x^2 + 5x + 6 \). For a quadratic function in the form \( y = ax^2 + bx + c \), the coefficient \( a \) determines the direction the parabola opens. Here, \( a = 1 \), which is positive. A positive \( a \) means the parabola opens upward.

Step2: Determine maximum or minimum

A parabola that opens upward has a minimum point (vertex) at the bottom, and no maximum (it extends upward infinitely). A parabola opening downward (negative \( a \)) has a maximum. Since \( a = 1 > 0 \), the graph has a minimum. Also, to find the vertex (or check the graph's x - intercepts for matching), we can factor \( y = x^2 + 5x + 6=(x + 2)(x + 3) \), so the roots are \( x=-2 \) and \( x = - 3 \). The vertex of \( y=ax^{2}+bx + c \) is at \( x=-\frac{b}{2a} \), for \( y=x^{2}+5x + 6 \), \( x =-\frac{5}{2\times1}=-2.5 \). The parabola opening upward with roots at - 3 and - 2 (close to - 2.5) will have a minimum at the vertex. The third graph (with the label "The graph has a minimum") should match the shape (opening upward, vertex between - 3 and - 2? Wait, no, wait, factoring gives roots at - 2 and - 3, so the vertex is at \( x=-2.5 \), so the graph with the minimum should have the vertex around \( x=-2.5 \)? Wait, maybe the options: the first two say "maximum" (wrong, since \( a>0 \)), the third says "minimum". So the correct one is the graph labeled "The graph has a minimum".

Answer:

The graph labeled "The graph has a minimum" (the third option in the given choices, with the parabola opening upward and vertex around the correct x - value for \( y = x^2+5x + 6 \)) is correct, and the graph has a minimum.