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.