Answer: We can determine which graph represents which polynomial function by analyzing the factors of each function.
First, let's consider f(x) = (x + 3)(x + 4). The factors are (x + 3) and (x + 4). When we multiply these factors together, we get a quadratic polynomial with a positive leading coefficient. This means that the graph of f(x) will be a parabola that opens upward.
Next, let's consider g(x) = 1/3(x + 3)(x − 4). The factors are (x + 3) and (x - 4). When we multiply these factors together and simplify, we get a quadratic polynomial with a leading coefficient of 1/3. This means that the graph of g(x) will also be a parabola that opens upward, but it will be narrower than the graph of f(x).
Based on this analysis, we can determine that the graph of f(x) corresponds to the wider parabola, and the graph of g(x) corresponds to the narrower parabola. We can also determine this without using graphing software by noting that f(x) has roots at x = -3 and x = -4, while g(x) has roots at x = -3 and x = 4. The graph of f(x) must therefore intersect the x-axis at -3 and -4, while the graph of g(x) must intersect the x-axis at -3 and 4. By examining the graphs, we can see that the wider parabola intersects the x-axis at -3 and -4, so it corresponds to f(x), while the narrower parabola intersects the x-axis at -3 and 4, so it corresponds to g(x).
Explanation: