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The image shows a quadratic equation \( y = 4x^2 + x - 14 \) and a diagram of its graph. The graph cuts the x-axis at points A and B, and intersects the y-axis at point C. The problem asks for:

1. The coordinates of points A and C.
2. The equation of the line of symmetry.

### Part (i) - Coordinates of Points A and C
- **To find point A and B (x-intercepts):** Set \( y = 0 \) and solve the quadratic equation:
  \[
  4x^2 + x - 14 = 0
  \]
  We can solve this using the quadratic formula:
  \[
  x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
  \]
  where \( a = 4 \), \( b = 1 \), and \( c = -14 \).

- **To find point C (y-intercept):** Set \( x = 0 \) and solve for \( y \).

### Part (ii) - Equation of the Line of Symmetry
The line of symmetry for a quadratic function \( y = ax^2 + bx + c \) is given by:
\[
x = \frac{-b}{2a}
\]

I will calculate both the coordinates of points A and C, as well as the equation of the line of symmetry. Let's begin the calculations.### Solutions:

1. **Coordinates of Points A and B (x-intercepts):**
   - \( A(1.75, 0) \)
   - \( B(-2, 0) \)

2. **Coordinates of Point C (y-intercept):**
   - \( C(0, -14) \)

3. **Equation of the Line of Symmetry:**
   - \( x = -0.125 \)

Would you like further details on any step, or do you have any other questions?

Here are 5 related questions to explore further:
1. How do you derive the quadratic formula?
2. What is the significance of the discriminant in quadratic equations?
3. How do you graph a quadratic equation by hand?
4. Can the vertex of a parabola be used to find the maximum or minimum value?
5. How does completing the square help in solving quadratic equations?

**Tip:** When solving quadratic equations, always check the discriminant to know the nature of the roots.

The diagram shows the graph of y = 4x^2 + x - 14. It cuts the x-axis at two points A and B, and the y-axis at the point C. Find: (i) the coordinates of A and C, (ii) the equation of the line of symmetry.

This problem involves solving the quadratic equation y = 4x^2 + x - 14 by finding the x-intercepts (points A and B), the y-intercept (point C), and the equation of the line of symmetry. Learn how to use the quadratic formula and identify key properties of the parabola.

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