To determine which graph represents the equation [tex]\( y = \frac{3}{2} x^2 - 6x \)[/tex], we need to go through a few steps to analyze the given quadratic function.
### Step-by-Step Solution:
1. Standard Form of the Equation:
The given equation is [tex]\( y = \frac{3}{2} x^2 - 6x \)[/tex]. This is a quadratic equation in the standard form [tex]\( y = ax^2 + bx + c \)[/tex], where [tex]\( a = \frac{3}{2} \)[/tex], [tex]\( b = -6 \)[/tex], and [tex]\( c = 0 \)[/tex].
2. Determine the Vertex:
The vertex form of a quadratic equation is [tex]\( y = a(x - h)^2 + k \)[/tex], where [tex]\( (h, k) \)[/tex] is the vertex. To find the vertex from the standard form, we use:
[tex]\[
h = -\frac{b}{2a}
\][/tex]
Substituting [tex]\( a = \frac{3}{2} \)[/tex] and [tex]\( b = -6 \)[/tex]:
[tex]\[
h = -\frac{-6}{2 \times \frac{3}{2}} = \frac{6}{3} = 2
\][/tex]
Now we find [tex]\( k \)[/tex] by substituting [tex]\( h = 2 \)[/tex] back into the original equation:
[tex]\[
k = \frac{3}{2}(2)^2 - 6(2) = \frac{3}{2} \times 4 - 12 = 6 - 12 = -6
\][/tex]
So, the vertex is [tex]\( (2, -6) \)[/tex].
3. Determine Concavity:
The sign of [tex]\( a \)[/tex] determines the direction of the parabola. Since [tex]\( a = \frac{3}{2} \)[/tex] (which is positive), the parabola opens upwards.
4. X-Intercepts (Roots):
To find the x-intercepts, we set [tex]\( y = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[
0 = \frac{3}{2}x^2 - 6x
\][/tex]
Factoring out [tex]\( \frac{3}{2}x \)[/tex]:
[tex]\[
\frac{3}{2} x (x - 4) = 0
\][/tex]
So, the solutions are [tex]\( x = 0 \)[/tex] and [tex]\( x = 4 \)[/tex]. Hence, the x-intercepts are [tex]\( (0, 0) \)[/tex] and [tex]\( (4, 0) \)[/tex].
5. Y-Intercept:
The y-intercept occurs when [tex]\( x = 0 \)[/tex]. Substituting [tex]\( x = 0 \)[/tex] into the equation gives:
[tex]\[
y = \frac{3}{2} (0)^2 - 6 (0) = 0
\][/tex]
So, the y-intercept is [tex]\( (0, 0) \)[/tex].
### Summary of Key Points:
- The parabola opens upwards.
- The vertex is [tex]\( (2, -6) \)[/tex].
- The x-intercepts are at [tex]\( (0, 0) \)[/tex] and [tex]\( (4, 0) \)[/tex].
- The y-intercept is at [tex]\( (0, 0) \)[/tex].
Based on these characteristics, we should choose the graph which matches these points and features.
