Question

graph yequals4xcubednegative 6xsquared    

Answer 1

To graph the function \( y = 4x^3 - 6x^2 \), we start by analyzing and sketching its key components, including its form, intercepts, critical points, and overall behavior.

### Step 1: Identify the function characteristics

1. **Form**: This is a cubic polynomial. Its general shape will be determined by its leading term, \( 4x^3 \).
2. **Intercepts**:
- **Y-Intercept**: Set \( x = 0 \):
\[
y = 4(0)^3 - 6(0)^2 = 0
\]
So the y-intercept is \( (0, 0) \).

- **X-Intercepts**: Set \( y = 0 \):
\[
4x^3 - 6x^2 = 0
\]
Factor out \( 2x^2 \):
\[
2x^2(2x - 3) = 0
\]
This gives the solutions:
\[
2x^2 = 0 \quad \Rightarrow \quad x = 0 \quad \text{(double root)}
\]
\[
2x - 3 = 0 \quad \Rightarrow \quad x = \frac{3}{2}
\]
So, the x-intercepts are \( (0, 0) \) and \( \left( \frac{3}{2}, 0 \right) \).

### Step 2: Find critical points

To find critical points, we take the derivative \( y' \) and set it to zero:
\[
y' = 12x^2 - 12x
\]
Factoring gives:
\[
y' = 12x(x - 1) = 0
\]
Setting this to zero:
- \( x = 0 \)
- \( x = 1 \)

### Step 3: Determine the nature of critical points

To determine whether these critical points are maxima, minima, or points of inflection, we can use the second derivative:
\[
y'' = 24x - 12
\]
- At \( x = 0 \):
\[
y''(0) = 24(0) - 12 = -12 \quad \text{(local maximum)}
\]
- At \( x = 1 \):
\[
y''(1) = 24(1) - 12 = 12 \quad \text{(local minimum)}
\]

### Step 4: Calculate values at critical points

Now we can find the corresponding \( y \) values:
- At \( x = 0 \):
\[
y(0) = 0
\]
- At \( x = 1 \):
\[
y(1) = 4(1)^3 - 6(1)^2 = 4 - 6 = -2
\]

### Step 5: Analyze end behavior

Since it is a cubic polynomial, as \( x \to -\infty \), \( y \to -\infty \) and as \( x \to +\infty \), \( y \to +\infty \).

### Step 6: Sketch the graph

Using the information gathered, we can now sketch the graph:

- X-Intercepts: \( (0, 0) \) and \( \left( \frac{3}{2}, 0 \right) \)
- Critical points: Local maximum at \( (0,0) \) and local minimum at \( (1, -2) \)
- End behavior: Down left, up right.

The graph will start from the lower left, rise through the origin, reach a local maximum at (0, 0), then drop down to a local minimum at (1, -2), and finally rise again through \( \left( \frac{3}{2}, 0 \right) \) heading up towards positive infinity on the right.

You can plot these points and features on graph paper or using graphing software for a clearer representation.

If you have specific questions or need further assistance with this function, feel free to ask!