This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Hey Trinity🦋💓 — ready when you are. Here's how to solve the problem: b) (i) Equation of the graph Step 1: Use the x-intercepts to form the factored equation. The graph meets the x-axis at x = -4 and x = 2. These are the roots of the quadratic equation. The general factored form of a quadratic equation is y = a(x - x_1)(x - x_2), where x_1 and x_2 are the roots. Substituting the given roots: y = a(x - (-4))(x - 2) y = a(x + 4)(x - 2) Step 2: Find the value of 'a' using the y-intercept. From the graph, the parabola crosses the y-axis (where x=0) at y = -8. This is the point (0, -8). Substitute this point into the equation from Step 1: -8 = a(0 + 4)(0 - 2) -8 = a(4)(-2) -8 = -8a a = 1 Step 3: Write the final equation by expanding. Substitute a=1 back into the factored form and expand: y = 1(x + 4)(x - 2) y = x^2 - 2x + 4x - 8 y = x^2 + 2x - 8 The equation of the graph is y = x^2 + 2x - 8. b) (ii) Coordinates of the turning point Step 1: Find the x-coordinate of the turning point. For a quadratic equation in the form y = ax^2 + bx + c, the x-coordinate of the turning point (vertex) is given by the formula x = (-b)/(2a). From our equation y = x^2 + 2x - 8, we have a=1 and b=2. x = (-2)/(2(1)) x = -1 Step 2: Find the y-coordinate of the turning point. Substitute the x-coordinate (x = -1) back into the equation of the graph: y = (-1)^2 + 2(-1) - 8 y = 1 - 2 - 8 y = -1 - 8 y = -9 The coordinates of the turning point are (-1, -9). Drop the next question 📸