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We are given the equation of the curve:

\[
y = (m - 2)x^2 + 2mx + (m - 3)
\]

and it is stated that the graph intersects the x-axis, which means \(y = 0\) at the x-intercepts. Let's go step by step.

### Part (a) 
**Find the smallest integral value of \(m\).**

For the graph to intersect the x-axis, we solve for when \(y = 0\). Thus, the equation becomes:

\[
(m - 2)x^2 + 2mx + (m - 3) = 0
\]

This is a quadratic equation in the form of \(Ax^2 + Bx + C = 0\). The condition for the equation to have real x-intercepts is that the discriminant must be non-negative. The discriminant \(\Delta\) for a quadratic equation \(Ax^2 + Bx + C = 0\) is given by:

\[
\Delta = B^2 - 4AC
\]

In this case:
- \(A = m - 2\)
- \(B = 2m\)
- \(C = m - 3\)

Substituting these into the discriminant formula:

\[
\Delta = (2m)^2 - 4(m - 2)(m - 3)
\]
\[
\Delta = 4m^2 - 4[(m - 2)(m - 3)]
\]
\[
\Delta = 4m^2 - 4[m^2 - 5m + 6]
\]
\[
\Delta = 4m^2 - 4m^2 + 20m - 24
\]
\[
\Delta = 20m - 24
\]

For the graph to intersect the x-axis, the discriminant must be non-negative:

\[
20m - 24 \geq 0
\]
\[
20m \geq 24
\]
\[
m \geq \frac{24}{20} = 1.2
\]

Since \(m\) must be an integer and \(m \neq 2\), the smallest integer value for \(m\) is **\(m = 2\)**.

However, the problem explicitly states that \(m \neq 2\), so the next smallest integer is **\(m = 3\)**.

### Part (b)
**When \(m = 3\), does the graph pass through the origin?**

To determine if the graph passes through the origin when \(m = 3\), substitute \(m = 3\) into the equation:

\[
y = (3 - 2)x^2 + 2(3)x + (3 - 3)
\]
\[
y = x^2 + 6x
\]

Now, check whether the graph passes through the origin by substituting \(x = 0\) into the equation:

\[
y = 0^2 + 6(0) = 0
\]

Therefore, when \(m = 3\), the graph **does pass through the origin**.

---

Would you like more details on any step? Here are five related questions to extend the learning:

1. How does the discriminant affect the number of x-intercepts for a quadratic equation?
2. What happens if the discriminant is exactly zero?
3. Could we use another method to determine if the graph passes through the origin without substituting values?
4. How would the graph look if \(m = 2\), despite the constraint \(m \neq 2\)?
5. How do the coefficients of \(x^2\) and \(x\) affect the shape of the parabola?

**Tip:** Always check the discriminant first when determining if a quadratic equation has real roots (x-intercepts).

It is given that the graph of y = (m - 2)x^2 + 2mx + (m - 3) intersects the x-axis, where m ≠ 2.
(a) Find the smallest integral value of m.
(b) When m takes the value in (a), does the graph pass through the origin? Explain your answer.

This problem involves solving a quadratic equation to determine the smallest integral value of m such that the graph intersects the x-axis and determining if the graph passes through the origin. Learn how to use the discriminant theorem and quadratic formulas.

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