Question

use your graphing calculator to sketch the graph of the function, and then determine the x-interval over which the function is increasing.

g(x) = -x² + 4x + 5

sketch the graph of the function in the viewing window -10,10×-10,10. choose the correct graph below.
○ a. ○ b. ○ c. ○ d.
(graphs are shown with options a, b, c, d, each with zoom and other icons)
find the x-interval, expressed as an inequality, over which the function is increasing.
(type an inequality or a compound inequality.)

Explanation:

Step1: Identify the function type

The function \(g(x)=-x^{2}+4x + 5\) is a quadratic function. The general form of a quadratic function is \(y = ax^{2}+bx + c\). Here, \(a=- 1\), \(b = 4\), \(c = 5\). Since \(a=-1<0\), the parabola opens downwards.

Step2: Find the vertex of the parabola

The x - coordinate of the vertex of a parabola given by \(y=ax^{2}+bx + c\) is \(x=-\frac{b}{2a}\). Substitute \(a=-1\) and \(b = 4\) into the formula:
\(x=-\frac{4}{2\times(-1)}=-\frac{4}{-2} = 2\)

Step3: Determine the increasing interval

For a parabola that opens downwards (because \(a<0\)), the function is increasing to the left of the vertex and decreasing to the right of the vertex. So the function \(g(x)\) is increasing when \(x<2\).

Answer:

\(x < 2\) (or in interval notation, \((-\infty, 2)\))