Question

which equation best matches the graph shown below?
answer
\\( y = 3(x + 4)(x + 2) \\) \\( y = 3(x - 4)(x - 2) \\)
\\( y = -3(x - 4)(x - 2) \\) \\( y = -3(x + 4)(x + 2) \\)

Explanation:

Step1: Identify x-intercepts

The graph crosses the x-axis at \( x = 2 \) and \( x = 4 \). So the roots are \( x = 2 \) and \( x = 4 \), which means the factored form should be \( y = a(x - 2)(x - 4) \) (since if \( x = r \) is a root, the factor is \( (x - r) \)).

Step2: Determine the leading coefficient sign

The parabola opens downward, so the leading coefficient \( a \) must be negative.

Step3: Match with options

Looking at the options, \( y = -3(x - 4)(x - 2) \) has roots at \( x = 4 \) and \( x = 2 \) (since \( x - 4 = 0 \) gives \( x = 4 \), \( x - 2 = 0 \) gives \( x = 2 \)) and the coefficient \( -3 \) is negative (so it opens downward), matching the graph.

Answer:

\( y = -3(x - 4)(x - 2) \) (the third option: \( y = -3(x - 4)(x - 2) \))