Question

use your graphing calculator to sketch the graph of the function, and then determine the x-interval over which the function is decreasing.\\( g(x) = -x^2 + 13x - 40 \\)\\( \dots \\)sketch the graph of the function in the viewing window \\( -10,10 \times -10,10 \\). choose the correct graph below.\\( \circ \\) a.\\( \circ \\) b.\\( \circ \\) c.\\( \circ \\) d.

Explanation:

Step 1: Analyze the function's form

The function $g(x)=-x^{2}+13x - 40$ is a quadratic function in the form $y = ax^{2}+bx + c$, where $a=- 1$, $b = 13$, and $c=-40$. Since $a=-1<0$, the parabola opens downwards.

Step 2: Find the vertex of the parabola

The x - coordinate of the vertex of a quadratic function $y=ax^{2}+bx + c$ is given by $x=-\frac{b}{2a}$. Substituting $a=-1$ and $b = 13$ into the formula, we get $x=-\frac{13}{2\times(-1)}=\frac{13}{2}=6.5$.
For a downward - opening parabola, the function is increasing to the left of the vertex and decreasing to the right of the vertex. So the function $g(x)$ is decreasing for $x>\frac{13}{2}$, or in interval notation $(\frac{13}{2},\infty)$.

Step 3: Analyze the graphs

• Option A: The parabola opens upwards, so it can't be the graph of $g(x)$ (since $a=-1<0$ for $g(x)$).
• Option B: The parabola opens upwards, so it can't be the graph of $g(x)$.
• Option C: The parabola opens downwards. We can also check the roots of the function. To find the roots, set $g(x)=0$, so $-x^{2}+13x - 40 = 0$, or $x^{2}-13x + 40=0$. Factoring, we get $(x - 5)(x - 8)=0$, so the roots are $x = 5$ and $x = 8$. In the viewing window $[-10,10]\times[-10,10]$, a downward - opening parabola with roots at $x = 5$ and $x = 8$ is consistent with option C.
• Option D: The roots of the parabola seem to be outside the range that we calculated (roots at $x = 5$ and $x = 8$), so it is not the correct graph.

Answer:

C. (assuming the third graph is a downward - opening parabola with roots in a reasonable position; and the interval where the function is decreasing is $(\frac{13}{2},\infty)$ or $(6.5,\infty)$)