To determine which of the characteristics of the graph of the equation [tex]\(y = 2x^2 - 4x - 4 \)[/tex] is displayed as a constant or coefficient in the equation, we need to analyze the structure of the quadratic equation.
A quadratic equation is generally represented in the form:
[tex]\[ y = ax^2 + bx + c \][/tex]
Here, [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are constants or coefficients:
- [tex]\(a\)[/tex] is the coefficient of [tex]\(x^2\)[/tex]
- [tex]\(b\)[/tex] is the coefficient of [tex]\(x\)[/tex]
- [tex]\(c\)[/tex] is the constant term
First, let's understand what each term [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] represents in a quadratic equation:
1. The coefficient [tex]\(a\)[/tex]:
- The coefficient [tex]\(a\)[/tex] determines the direction of the parabola (upwards if [tex]\(a > 0\)[/tex] or downwards if [tex]\(a < 0\)[/tex]).
- It also affects the width of the parabola.
2. The coefficient [tex]\(b\)[/tex]:
- The coefficient [tex]\(b\)[/tex] affects the placement of the vertex but does not directly determine the vertex's coordinates in a simple manner.
3. The constant [tex]\(c\)[/tex]:
- The constant [tex]\(c\)[/tex] represents the value of [tex]\(y\)[/tex] when [tex]\(x = 0\)[/tex].
- This means [tex]\(c\)[/tex] is the [tex]\(y\)[/tex]-intercept of the graph.
To confirm this, consider the point where the graph intersects the y-axis. At this point, [tex]\(x = 0\)[/tex]. Substituting [tex]\(x = 0\)[/tex] into the equation [tex]\(y = 2x^2 - 4x - 4\)[/tex], we get:
[tex]\[ y = 2(0)^2 - 4(0) - 4 \][/tex]
[tex]\[ y = -4 \][/tex]
Therefore, the [tex]\(y\)[/tex]-intercept of the graph is [tex]\(-4\)[/tex]. This shows that the constant term [tex]\(c\)[/tex] in the quadratic equation [tex]\(y = 2x^2 - 4x - 4\)[/tex], which is [tex]\(-4\)[/tex], is indeed the [tex]\(y\)[/tex]-intercept of the graph.
Hence, the characteristic of the graph that is displayed as a constant or coefficient in the equation is the [tex]\(y\)[/tex]-intercept.
The correct answer is:
(D) [tex]\(y\)[/tex]-intercept
