Question

which is the graph of $f(x) = -(x + 3)(x + 1)$? four graphs with options below each

Explanation:

Step1: Find the roots of the function

To find the roots, set \( f(x) = 0 \):
\( 0 = -(x + 3)(x + 1) \)
This implies \( x + 3 = 0 \) or \( x + 1 = 0 \), so the roots are \( x = -3 \) and \( x = -1 \).

Step2: Determine the direction of the parabola

The function is in the form \( f(x) = - (x + 3)(x + 1) \). Expanding, we get \( f(x) = - (x^2 + 4x + 3) = -x^2 - 4x - 3 \). The coefficient of \( x^2 \) is \( -1 \), which is negative. So the parabola opens downward.

Step3: Analyze the graphs

• The first graph has roots around positive values, which doesn't match \( x = -3 \) and \( x = -1 \).
• The second graph has roots at \( x = -3 \) and \( x = -1 \) (since it crosses the x - axis at these negative values) and opens downward (since the parabola opens down as the coefficient of \( x^2 \) is negative).
• The third and fourth graphs open upward (since their parabolas open up, as the coefficient of \( x^2 \) would be positive for them), which does not match our function.

Answer:

The second graph (the one with roots at \( x=-3 \) and \( x = - 1 \) and opening downward)