Which Graph Matches the Equation y = 3/2 x³?
Introduction
If you are looking for the graph that matches the equation y = 3/2 x³, you are dealing with a cubic function that has been vertically stretched by a factor of 1.In real terms, 5. This article will walk you through every step needed to identify the correct graph, explain the underlying mathematics, and provide a clear visual description that you can use to spot the right picture among multiple‑choice options. By the end, you will understand exactly how the coefficient, the exponent, and the shape of the curve interact, enabling you to answer the question confidently and accurately.
Understanding the Equation
The expression y = 3/2 x³ can be broken down into three distinct parts:
- Coefficient (3/2) – This is a positive fraction equal to 1.5. It tells us how much the graph is stretched vertically compared to the basic cubic y = x³.
- Exponent (³) – The cube indicates an odd power. Cubic functions are symmetric about the origin (they are odd functions), meaning the left side of the graph mirrors the right side when rotated 180°.
- Variable (x) – The variable appears alone, so there are no horizontal shifts or additional transformations.
Because the coefficient is positive, the graph will rise to the right (as x increases, y becomes increasingly positive) and fall to the left (as x becomes more negative, y turns increasingly negative). The steepness of the curve is greater than that of y = x³, since each y‑value is multiplied by 1.5 No workaround needed..
Key Features of the Graph
1. Shape – Cubic Curve
- Smooth, continuous line with no breaks.
- S‑shaped turning point at the origin (0, 0).
- The curve is monotonic (always increasing) because the derivative, dy/dx = (9/2) x², is always non‑negative and only zero at x = 0.
2. Intercepts
- Y‑intercept: (0, 0) – the graph passes through the origin.
- X‑intercepts: Only at x = 0, because the equation y = (3/2) x³ equals zero only when x = 0.
3. Points to Plot for Visual Confirmation
| x | y = (3/2) x³ | Description |
|---|---|---|
| -2 | -12 | Steep negative side |
| -1 | -1.5 | Slightly negative |
| 0 | 0 | Origin |
| 1 | 1.5 | Slightly positive |
| 2 | 12 | Steep positive side |
Plotting these points will give you a clear picture of how the curve behaves. Notice the symmetry: the point at x = -1 (‑1.5) mirrors the point at x = 1 (1.5).
4. Vertical Stretch
If you compare y = (3/2) x³ with the parent function y = x³, every y‑value is multiplied by 1.5. This means:
- At x = 1, y = 1.5 instead of 1.
- At x = 2, y = 12 instead of 8.
The graph appears “taller” or “steeper” while preserving the overall cubic shape Nothing fancy..
Plotting the Curve Step‑by‑Step
- Draw the axes and label the origin.
- Mark the key points from the table above.
- Connect the points with a smooth, flowing line, ensuring the curve passes through the origin and maintains the same direction on both sides of the y‑axis.
- Check the end behavior: as x → ∞, y → ∞; as x → –∞, y → –∞.
When you sketch the graph, you will see a single S‑shaped curve that is steeper than the basic y = x³ but still symmetric about the origin.
Comparing with Common Graph Options
When a multiple‑choice question asks which graph matches the equation y = 3/2 x³, the distractors usually fall into one of these categories:
| Distractor Type | Typical Mistake | Why It’s Wrong |
|---|---|---|
| Vertical compression | y = (2/3) x³ | Coefficient < 1 makes the curve flatter, not steeper. |
| Horizontal shift | y = (3/2) (x‑2)³ | Adding a term inside the parentheses moves the graph left/right, breaking origin symmetry. |
| Reflection | y = -(3/2) x³ | Negative coefficient flips the graph vertically, causing it to fall to the right instead of rising. |
| Different exponent | y = (3/2) x² | Quadratic shape is parabolic, not cubic; it lacks the S‑shape and end‑behavior. |
The correct graph will exhibit the steep, S‑shaped curve passing through the origin, with points like (2, 12) and (‑2, ‑12) clearly plotted Turns out it matters..
Scientific Explanation of the Cubic Shape
A cubic function’s derivative is a quadratic:
[ \frac{dy}{dx} = \frac{3\cdot 3}{2}x^{2} = \frac{9}{2}x^{2} ]
Since (x^{2}) is always non‑negative, the slope is zero only at x = 0 and positive everywhere else. This tells us:
- The graph is flat at the origin (horizontal tangent).
- As x moves away from zero, the slope grows proportionally to (x^{2}), causing the curve to become increasingly steep.
The second derivative, ( \frac{d^{
The second derivative, ( \frac{d^{2}y}{dx^{2}} = 9x ), changes sign at ( x = 0 ). This confirms that the origin is an inflection point—the precise location where the curve transitions from concave down (for ( x < 0 )) to concave up (for ( x > 0 )). Unlike quadratic functions, which maintain a single concavity, this shift in curvature is the defining geometric signature of any cubic polynomial with a non-zero quadratic term in its derivative. The positive leading coefficient (( \frac{3}{2} > 0 )) guarantees the "left-to-right" upward orientation, ensuring the graph enters the third quadrant and exits the first.
Practical Verification Checklist
Before finalizing your selection on a test or assignment, run the candidate graph through this quick mental checklist:
- Origin Test: Does the curve pass exactly through (0,0)?
- Quadrant Check: Is the graph in Quadrant III (negative x, negative y) and Quadrant I (positive x, positive y) only? (No parts in Quadrants II or IV).
- Steepness Benchmark: Does the point (2, 12) lie on the curve? If the grid marks are standard, this point should sit noticeably higher than the parent function’s (2, 8).
- Flatness at Center: Does the curve appear to "flatten out" momentarily as it crosses the origin, rather than crossing at a sharp 45-degree angle?
- Symmetry: If you rotate the graph 180° about the origin, does it map perfectly onto itself? (Odd function symmetry).
If the graph passes all five checks, you have identified the correct representation of ( y = \frac{3}{2}x^{3} ).
Conclusion
Graphing ( y = \frac{3}{2}x^{3} ) is an exercise in recognizing how a single coefficient transforms a parent function. By anchoring your sketch to a few calculated coordinate pairs (( \pm1, \pm1.In practice, the multiplier ( \frac{3}{2} ) acts as a vertical stretch factor, amplifying every output value by 50% without altering the fundamental cubic nature—its odd symmetry, its single inflection point at the origin, or its opposing end behaviors. 5 ) and ( \pm2, \pm12 )) and verifying the calculus-driven concavity shift, you move beyond rote memorization to structural understanding. Whether you are matching a multiple-choice option or sketching by hand, the "steeper S-curve through the origin" remains the unmistakable visual fingerprint of this specific cubic equation Nothing fancy..
