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
Here's the solution for question 2a: To sketch the graph of y = -x^2+4x+1, we need to find its key features: the vertex, y-intercept, and x-intercepts. Step 1: Determine the direction of the parabola. The coefficient of x^2 is -1, which is negative. Therefore, the parabola opens downwards. Step 2: Find the vertex of the parabola. For a quadratic function y = ax^2+bx+c, the x-coordinate of the vertex is given by x = -(b)/(2a). Here, a=-1, b=4, c=1. x = -(4)/(2(-1)) = -(4)/(-2) = 2 Substitute x=2 into the equation to find the y-coordinate: y = -(2)^2 + 4(2) + 1 = -4 + 8 + 1 = 5 The vertex is at (2, 5). This is the maximum point of the parabola. Step 3: Find the y-intercept. The y-intercept occurs when x=0. y = -(0)^2 + 4(0) + 1 = 1 The y-intercept is at (0, 1). Step 4: Find the x-intercepts. The x-intercepts occur when y=0. -x^2+4x+1 = 0 Multiply by -1: x^2-4x-1 = 0 Use the quadratic formula x = -b ± sqrt(b^2-4ac)2a: x = -(-4) ± sqrt((-4)^2 - 4(1)(-1))2(1) x = 4 ± sqrt(16 + 4)2 x = 4 ± sqrt(20)2 x = 4 ± 2sqrt(5)2 x = 2 ± sqrt(5) The x-intercepts are approximately (2+sqrt(5), 0) ≈ (4.24, 0) and (2-sqrt(5), 0) ≈ (-0.24, 0). Step 5: Sketch the graph. • Plot the vertex (2, 5). • Plot the y-intercept (0, 1). • Plot the x-intercepts (2+sqrt(5), 0) and (2-sqrt(5), 0). • Draw a smooth parabola opening downwards, passing through these points. The axis of symmetry is the vertical line x=2. The key points for the sketch are: • Vertex: (2, 5) • Y-intercept: (0, 1) • X-intercepts: (2-sqrt(5), 0) and (2+sqrt(5), 0) Last free one today — make it count tomorrow, or type /upgrade for unlimited.