Question

use your graphing calculator to sketch the graph of the following quadratic function, then determine the domain and range.

$g(x) = -x^2 + 4x + 45$

sketch the graph of the function in the viewing window $-10,10 \times -50,50$. choose the correct graph below.

\\(\bigcirc\\) a. \\(\bigcirc\\) b. \\(\bigcirc\\) c. \\(\bigcirc\\) d.

Explanation:

Step1: Analyze the quadratic function

The function is \( g(x)= -x^{2}+4x + 45\). The coefficient of \(x^{2}\) is \(- 1<0\), so the parabola opens downward.

Step2: Find the vertex

The x - coordinate of the vertex of a quadratic function \(y = ax^{2}+bx + c\) is given by \(x=-\frac{b}{2a}\). For \(g(x)=-x^{2}+4x + 45\), \(a=-1\) and \(b = 4\). So \(x=-\frac{4}{2\times(-1)}=\frac{-4}{-2} = 2\).
Substitute \(x = 2\) into the function: \(g(2)=-(2)^{2}+4\times2 + 45=-4 + 8+45=49\). So the vertex is at \((2,49)\).

Step3: Analyze the roots

Set \(g(x)=0\), then \(-x^{2}+4x + 45 = 0\), or \(x^{2}-4x - 45=0\). Factor the quadratic: \(x^{2}-4x - 45=(x - 9)(x + 5)=0\). So the roots are \(x = 9\) and \(x=-5\).

Step4: Match with the graphs

Since the parabola opens downward (because \(a=-1<0\)) and has roots at \(x=-5\) and \(x = 9\) and vertex at \((2,49)\), we look for the graph that opens downward, has a vertex at \(x = 2\) and crosses the x - axis at \(x=-5\) and \(x = 9\). Among the options, graph D satisfies these conditions.

Answer:

D