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
Fresh day Ignatius, let's solve. The problem asks to sketch the curve for part (a), which is y = x^2 - 6x. To sketch a curve, we identify its key features such as its shape, intercepts, and vertex. Step 1: Identify the type of curve and its orientation. The equation y = x^2 - 6x is a quadratic function of the form y = ax^2 + bx + c. This means the curve is a parabola. Since the coefficient of x^2 is a=1 (which is positive), the parabola opens upwards. Step 2: Find the x-intercepts. To find the x-intercepts, set y=0: x^2 - 6x = 0 Factor out x: x(x - 6) = 0 This gives two x-intercepts: x = 0 or x - 6 = 0 x = 6 The x-intercepts are (0, 0) and (6, 0). Step 3: Find the y-intercept. To find the y-intercept, set x=0: y = (0)^2 - 6(0) y = 0 The y-intercept is (0, 0). Step 4: Find the vertex. The x-coordinate of the vertex of a parabola y = ax^2 + bx + c is given by the formula x = -(b)/(2a). For y = x^2 - 6x, we have a=1 and b=-6. x = -(-6)/(2(1)) = (6)/(2) = 3 Now, substitute x=3 back into the original equation to find the y-coordinate of the vertex: y = (3)^2 - 6(3) y = 9 - 18 y = -9 The vertex of the parabola is (3, -9). The key features for sketching the curve y = x^2 - 6x are: • Type of curve: Parabola opening upwards. • x-intercepts: (0, 0) and (6, 0). • y-intercept: (0, 0). • Vertex: (3, -9). Send me the next one 📸