To solve this problem, we need to analyze the equation [tex]\( y = (x-2)(x-4) \)[/tex] and understand the characteristics it exhibits in the [tex]\( xy \)[/tex]-plane.
First, let's find out the [tex]\( x \)[/tex]-intercepts of the graph. The [tex]\( x \)[/tex]-intercepts are the points where the graph crosses the [tex]\( x \)[/tex]-axis, i.e., where [tex]\( y = 0 \)[/tex].
Given the equation [tex]\( y = (x-2)(x-4) \)[/tex], we set [tex]\( y \)[/tex] to 0 to find the [tex]\( x \)[/tex]-intercepts:
[tex]\[ 0 = (x-2)(x-4). \][/tex]
To find the solutions for [tex]\( x \)[/tex], we can set each factor equal to zero:
[tex]\[ x - 2 = 0 \quad \text{or} \quad x - 4 = 0. \][/tex]
Solving these equations, we get:
[tex]\[ x = 2 \quad \text{or} \quad x = 4. \][/tex]
So, the [tex]\( x \)[/tex]-intercepts of the graph are [tex]\( x = 2 \)[/tex] and [tex]\( x = 4 \)[/tex].
Now we need to determine which of the following characteristics of the graph is displayed as a constant or coefficient in the equation [tex]\( y = (x-2)(x-4) \)[/tex]:
(A) [tex]\( x \)[/tex]-coordinate of the vertex
(B) [tex]\( y \)[/tex]-coordinate of the vertex
(C) [tex]\( x \)[/tex]-intercept(s)
(D) [tex]\( y \)[/tex]-intercept
In this case, the equation shows that the graph crosses the [tex]\( x \)[/tex]-axis at [tex]\( x = 2 \)[/tex] and [tex]\( x = 4 \)[/tex]. Therefore, the [tex]\( x \)[/tex]-intercepts are directly revealed by the constants in the equation after setting [tex]\( y \)[/tex] to 0 and solving for [tex]\( x \)[/tex].
Thus, the characteristic displayed as a constant or coefficient in the equation is:
(C) [tex]\( x \)[/tex]-intercept(s)
