Question

use your graphing calculator to sketch the graph of the function, and then determine the coordinates of the x-intercepts for the function, if they exist.\\( g(x) = -x^2 + 11x - 18 \\)\\( \\)sketch the graph of the function in the viewing window \\( -10,10 \times -15,15 \\). choose the correct graph below\\( \\)\\( \bigcirc \\) a.\\( \bigcirc \\) b.\\( \bigcirc \\) c.\\( \bigcirc \\) d.

Explanation:

Step1: Analyze the function's leading coefficient

The function is \( g(x) = -x^2 + 11x - 18 \). The leading coefficient (of \( x^2 \)) is \(-1\), which is negative. So the parabola opens downward. This eliminates options A and C (which open upward, as they are U - shaped).

Step2: Check the remaining options (B and D)

Now, we can also find the x - intercepts by setting \( g(x)=0 \), i.e., \(-x^2 + 11x - 18 = 0\). Multiply both sides by - 1: \( x^2 - 11x + 18 = 0 \). Factor the quadratic: \( (x - 2)(x - 9)=0 \). So the x - intercepts are at \( x = 2 \) and \( x = 9 \). The graph should cross the x - axis at \( x = 2 \) and \( x = 9 \). Option B's graph (the middle - left among the remaining) shows a downward - opening parabola crossing the x - axis at two positive x - values, while option D seems to have a root at a non - 2 or 9 value (and the shape and intercepts match B better). Also, from the opening direction (downward) and the x - intercepts, option B is correct.

Answer:

B