On the axes, sketch the graph of y = (2x -5) (x+3) (1-x) stating the intercepts with the coordinate axes. 

To sketch the graph of the function \( y = (2x - 5)(x + 3)(1 - x) \), let's analyze the key points first, including the intercepts.

### Step 1: Find the x-intercepts
The x-intercepts occur when \( y = 0 \). Therefore, we set the equation equal to 0 and solve for \( x \):

\[
(2x - 5)(x + 3)(1 - x) = 0
\]

This gives us three factors:
- \( 2x - 5 = 0 \) ⇒ \( x = \frac{5}{2} \)
- \( x + 3 = 0 \) ⇒ \( x = -3 \)
- \( 1 - x = 0 \) ⇒ \( x = 1 \)

Thus, the x-intercepts are:
\[
x = \frac{5}{2}, \quad x = -3, \quad x = 1
\]

### Step 2: Find the y-intercept
The y-intercept occurs when \( x = 0 \). Substituting \( x = 0 \) into the equation:

\[
y = (2(0) - 5)(0 + 3)(1 - 0) = (-5)(3)(1) = -15
\]

Thus, the y-intercept is:
\[
y = -15
\]

### Step 3: Behavior of the graph
Now that we have the intercepts, let's analyze the behavior of the graph at these points:
- The polynomial is a cubic function (degree 3), so the graph will start from negative infinity, cross the x-axis at \( x = -3 \), \( x = 1 \), and \( x = \frac{5}{2} \), and end at positive infinity.
- The sign of the leading term (if expanded) is negative, so the graph starts in the negative direction.

### Summary of key points:
- **X-intercepts:** \( x = -3, 1, \frac{5}{2} \)
- **Y-intercept:** \( y = -15 \)

Would you like a sketch or details on how to expand the polynomial?

#### Related Questions:
1. How do you determine the end behavior of a cubic polynomial?
2. What are the general steps to find intercepts of polynomials?
3. How do turning points affect the graph of a polynomial?
4. What is the significance of the degree of a polynomial in its graph?
5. How do you find the critical points of a cubic function?

#### Tip:
When sketching polynomials, remember that their graphs cross the x-axis at real roots and touch but do not cross at repeated roots.

On the axes, sketch the graph of y = (2x - 5)(x + 3)(1 - x), stating the intercepts with the coordinate axes.

Learn how to sketch the graph of the cubic polynomial y = (2x - 5)(x + 3)(1 - x) by determining the x- and y-intercepts and analyzing the behavior of the function. This problem covers key algebraic concepts like polynomial roots, intercepts, and graph behavior. Suitable for students in Grades 10-12.

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