Identifying Graph of Quadratic Function
To solve the question given in the image, which is to identify the graph of the quadratic function g(x) = -x^2 - 7x - 12, we should look at the properties of the quadratic function and check which graph corresponds to these.
The given quadratic function is in the form g(x) = ax^2 + bx + c, where a = -1, b = -7, and c = -12. Because a is negative, we know that the parabola opens downward. The graph should therefore be a downward-opening parabola.
Now let's find the x-intercepts (the roots of the quadratic equation) by factoring or using the quadratic formula. The quadratic formula is given by x = [-b ± sqrt(b^2 - 4ac)] / 2a.
Using the coefficients from g(x) = -x^2 - 7x - 12 gives us:
x = [-(-7) ± sqrt((-7)^2 - 4 * (-1) * (-12))] / 2 * (-1)
x = [7 ± sqrt(49 - 48)] / -2
x = [7 ± sqrt(1)] / -2
x = [7 ± 1] / -2
This yields two solutions for x:
x = (7 + 1) / -2 = 8 / -2 = -4
x = (7 - 1) / -2 = 6 / -2 = -3
So the x-intercepts are x = -4 and x = -3.
By checking the options:
- Option A shows a parabola that does not cross the x-axis at -4 and -3.
- Option B shows a parabola with the correct roots of -4 and -3, but it is opening upward, which is incorrect.
- Option C shows a parabola that opens downwards, but with the incorrect roots.
- Option D shows a parabola opening downward with the correct roots of -4 and -3.
Thus, the correct graph that corresponds to the function g(x) = -x^2 - 7x - 12 is the one in Option D.
