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
To sketch the graph of y = 5 - 2x - x^2, we will find its key features: the direction of opening, the vertex, the y-intercept, and the x-intercepts. Step 1: Determine the direction of opening. The equation is y = -x^2 - 2x + 5. Since the coefficient of x^2 is -1 (which is negative), the parabola opens downwards. Step 2: Find the vertex. The x-coordinate of the vertex for a parabola y = ax^2 + bx + c is given by x = -(b)/(2a). For y = -x^2 - 2x + 5, we have a=-1 and b=-2. x = -(-2)/(2(-1)) = -(-2)/(-2) = -1 Now, substitute x=-1 into the equation to find the y-coordinate: y = 5 - 2(-1) - (-1)^2 y = 5 + 2 - 1 y = 6 The vertex is at (-1, 6). Step 3: Find the y-intercept. Set x=0 in the equation: y = 5 - 2(0) - (0)^2 y = 5 The y-intercept is at (0, 5). Step 4: Find the x-intercepts (roots). Set y=0 in the equation: 0 = 5 - 2x - x^2 Rearrange to x^2 + 2x - 5 = 0. Use the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a: x = -2 ± sqrt(2^2 - 4(1)(-5))2(1) x = -2 ± sqrt(4 + 20)2 x = -2 ± sqrt(24)2 x = -2 ± 2sqrt(6)2 x = -1 ± sqrt(6) The x-intercepts are approximately x_1 = -1 + sqrt(6) ≈ -1 + 2.45 = 1.45 and x_2 = -1 - sqrt(6) ≈ -1 - 2.45 = -3.45. So, the x-intercepts are approximately (1.45, 0) and (-3.45, 0). Step 5: Sketch the graph. Plot the vertex (-1, 6), the y-intercept (0, 5), and the x-intercepts (-1+sqrt(6), 0) and (-1-sqrt(6), 0). Draw a smooth parabolic curve opening downwards through these points. Summary of key points for sketching: • Opens downwards. • Vertex: (-1, 6) • Y-intercept: (0, 5) • X-intercepts: (-1+sqrt(6), 0) and (-1-sqrt(6), 0) (approximately (1.45, 0) and (-3.45, 0)) Drop the next question.