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The problem given involves a quadratic function \( g(x) = 2x^2 - mx + n \) which has a minimum point at \( \left( \frac{3}{4}, -\frac{49}{8} \right) \).

### Solution for Part (a):

To find the values of \( m \) and \( n \) using the completing the square method:

1. **Rewrite the quadratic function in the form**:  
   \[
   g(x) = 2x^2 - mx + n
   \]

2. **Complete the square** by expressing it in the form:  
   \[
   g(x) = a(x - h)^2 + k
   \]  
   where \( (h, k) \) is the vertex of the parabola. Here, \( h = \frac{3}{4} \) and \( k = -\frac{49}{8} \).

3. Since the coefficient of \( x^2 \) is 2, we can express the quadratic in terms of completing the square:  
   \[
   g(x) = 2 \left( x^2 - \frac{m}{2}x \right) + n
   \]

4. **Complete the square inside the parentheses**:  
   \[
   x^2 - \frac{m}{2}x = \left( x - \frac{m}{4} \right)^2 - \left( \frac{m}{4} \right)^2
   \]  
   So,  
   \[
   g(x) = 2 \left( \left( x - \frac{m}{4} \right)^2 - \frac{m^2}{16} \right) + n
   \]  
   \[
   g(x) = 2 \left( x - \frac{m}{4} \right)^2 - \frac{m^2}{8} + n
   \]

5. Now, since the vertex is given as \( \left( \frac{3}{4}, -\frac{49}{8} \right) \), we equate:  
   \[
   \frac{m}{4} = \frac{3}{4} \implies m = 3
   \]

6. Substitute \( m = 3 \) into the equation for \( g(x) \):  
   \[
   g(x) = 2 \left( x - \frac{3}{4} \right)^2 - \frac{3^2}{8} + n
   \]  
   \[
   g(x) = 2 \left( x - \frac{3}{4} \right)^2 - \frac{9}{8} + n
   \]

7. Since the minimum point has \( y = -\frac{49}{8} \), equate:  
   \[
   -\frac{9}{8} + n = -\frac{49}{8}
   \]  
   \[
   n = -\frac{49}{8} + \frac{9}{8} = -\frac{40}{8} = -5
   \]

Thus, the values are:
- \( m = 3 \)
- \( n = -5 \)

### Solution for Part (b):

To sketch the graph of \( g(x) = 2x^2 - 3x - 5 \) for \( -2 \leq x \leq 4 \):

1. **Find the vertex**: The vertex is already known from Part (a) to be \( \left( \frac{3}{4}, -\frac{49}{8} \right) \).

2. **Find the intercepts**:
   - **y-intercept**: Set \( x = 0 \):
     \[
     g(0) = 2(0)^2 - 3(0) - 5 = -5
     \]
     So, the y-intercept is at \( (0, -5) \).

   - **x-intercepts**: Set \( g(x) = 0 \):
     \[
     2x^2 - 3x - 5 = 0
     \]
     Solve for \( x \) using the quadratic formula:
     \[
     x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(-5)}}{2(2)}
     \]
     \[
     x = \frac{3 \pm \sqrt{9 + 40}}{4}
     \]
     \[
     x = \frac{3 \pm \sqrt{49}}{4}
     \]
     \[
     x = \frac{3 \pm 7}{4}
     \]
     \[
     x = \frac{10}{4} = 2.5, \quad x = \frac{-4}{4} = -1
     \]
     So, the x-intercepts are at \( x = -1 \) and \( x = 2.5 \).

3. **Sketch the graph**: Plot the vertex \( \left( \frac{3}{4}, -\frac{49}{8} \right) \), y-intercept \( (0, -5) \), and x-intercepts \( (-1, 0) \) and \( (2.5, 0) \). Draw the parabola opening upwards, passing through these points.

Would you like more details on any part of the solution? Here are some related questions:

1. What is the method of completing the square?
2. How do we find the vertex of a quadratic function using a formula?
3. How can we determine the axis of symmetry for a quadratic function?
4. Why does the quadratic function have a minimum point?
5. What are the properties of the vertex form of a quadratic function?

**Tip**: When sketching quadratic functions, always identify key points such as the vertex, axis of symmetry, and intercepts to understand the shape of the graph.

Diberi bahawa fungsi g(x) = 2x^2 - mx + n mempunyai titik minimum pada (3/4, -49/8). Given that the function g(x) = 2x^2 - mx + n has a minimum point at (3/4, -49/8). (a) Dengan menggunakan kaedah penyempurnaan kuasa dua, cari nilai m dan n. By using the completing the square method, find the value of m and of n. (b) Lakar graf g(x) bagi -2 ≤ x ≤ 4. Sketch graph of g(x) for -2 ≤ x ≤ 4.

Learn how to solve the quadratic function g(x) = 2x^2 - mx + n and find the values of m and n using the completing the square method. Also, sketch the graph of g(x) for the range -2 ≤ x ≤ 4. This solution involves understanding quadratic functions, the vertex form, and key concepts in graphing.

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