To determine which graph represents the equation [tex]\( y = \frac{3}{2} x^2 - 6x \)[/tex], we need to analyze the properties and shape of the graph.
### Step-by-Step Solution
1. Identify the Form of the Equation
- The given equation is a quadratic equation of the 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 Shape of the Graph
- Since [tex]\( a = \frac{3}{2} \)[/tex] is positive, the parabola opens upwards.
3. Find the Vertex of the Quadratic Function
- The vertex [tex]\( x \)[/tex]-coordinate of a quadratic function [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the formula [tex]\( x = -\frac{b}{2a} \)[/tex].
- Here, [tex]\( a = \frac{3}{2} \)[/tex] and [tex]\( b = -6 \)[/tex], so:
[tex]\[
x = -\frac{-6}{2 \cdot \frac{3}{2}} = \frac{6}{3} = 2
\][/tex]
- To find the [tex]\( y \)[/tex]-coordinate of the vertex, substitute [tex]\( x = 2 \)[/tex] back into the equation:
[tex]\[
y = \frac{3}{2} (2)^2 - 6 \cdot 2 = \frac{3}{2} \cdot 4 - 12 = 6 - 12 = -6
\][/tex]
- Therefore, the vertex of the parabola is at [tex]\( (2, -6) \)[/tex].
4. Find the Intercepts
- y-intercept: When [tex]\( x = 0 \)[/tex]:
[tex]\[
y = \frac{3}{2} (0)^2 - 6 \cdot 0 = 0
\][/tex]
- x-intercepts: Set [tex]\( y = 0 \)[/tex] and solve for [tex]\( x \)[/tex]:
[tex]\[
0 = \frac{3}{2} x^2 - 6x
\][/tex]
Factor the equation:
[tex]\[
0 = x \left( \frac{3}{2} x - 6 \right)
\][/tex]
Setting each factor to zero gives:
[tex]\[
x = 0 \quad \text{or} \quad \frac{3}{2} x - 6 = 0
\][/tex]
Solving [tex]\( \frac{3}{2} x - 6 = 0 \)[/tex]:
[tex]\[
\frac{3}{2} x = 6 \quad \Rightarrow \quad x = 4
\][/tex]
- Thus, the x-intercepts are [tex]\( x = 0 \)[/tex] and [tex]\( x = 4 \)[/tex].
5. Graph Characteristics
- The graph is a parabola that opens upwards.
- The vertex of the parabola is at [tex]\( (2, -6) \)[/tex].
- The y-intercept is at [tex]\( (0, 0) \)[/tex].
- The x-intercepts are at [tex]\( (0, 0) \)[/tex] and [tex]\( (4, 0) \)[/tex].
### Conclusion
Based on the properties and characteristics of the graph, you should examine the given options (A and B) to see which one displays these features:
- A parabola opening upwards.
- A vertex at [tex]\( (2, -6) \)[/tex].
- Intercepts at [tex]\( x = 0 \)[/tex] and [tex]\( x = 4 \)[/tex], and [tex]\( y = 0 \)[/tex].
By reviewing these criteria, you should be able to identify the correct graph that matches these properties.
