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
7.1) To sketch the graphs of f(x) = x^2 - 4 and g(x) = -2x - 4: For f(x) = x^2 - 4 (parabola): • Shape: Opens upwards. • y-intercept: Set x=0 f(0) = 0^2 - 4 = -4. Point: (0, -4). • x-intercepts: Set f(x)=0 x^2 - 4 = 0 (x-2)(x+2) = 0 x = 2 or x = -2. Points: (2, 0) and (-2, 0). • Turning point: For f(x) = ax^2 + bx + c, the x-coordinate of the turning point is x = (-b)/(2a). Here, a=1, b=0, so x = (0)/(2(1)) = 0. The y-coordinate is f(0) = -4. Point: (0, -4). For g(x) = -2x - 4 (straight line): • y-intercept: Set x=0 g(0) = -2(0) - 4 = -4. Point: (0, -4). • x-intercept: Set g(x)=0 -2x - 4 = 0 -2x = 4 x = -2. Point: (-2, 0). Sketching instructions: Draw a Cartesian plane with x and y axes. 1. Plot the points (0, -4), (2, 0), and (-2, 0) for f(x). Draw a smooth parabola passing through these points, opening upwards, with (0, -4) as its minimum turning point. 2. Plot the points (0, -4) and (-2, 0) for g(x). Draw a straight line passing through these two points. Notice that both graphs intersect at (0, -4) and (-2, 0). 7.2) Use your graphs to determine the values of x if: 7.2.1) f(x) = g(x) This occurs at the intersection points of the two graphs. Step 1: Set the equations equal to each other. x^2 - 4 = -2x - 4 Step 2: Rearrange the equation to solve for x. x^2 + 2x = 0 Step 3: Factor out x. x(x+2) = 0 Step 4: Solve for x. x = 0 or x = -2 The values of x where f(x) = g(x) are x = 0 or x = -2. 7.2.2) f(x) < 0 This means finding the x-values where the graph of f(x) is below the x-axis. Step 1: Identify the x-intercepts of f(x), which are x = -2 and x = 2. Step 2: Since the parabola f(x) = x^2 - 4 opens upwards, it is below the x-axis between its x-intercepts. The values of x where f(x) < 0 are -2 < x < 2. 7.3) How can you use the graph of g to obtain the graph of k(x) = -2x + 3? Given g(x) = -2x - 4. To obtain k(x) = -2x + 3 from g(x), we compare the two functions: k(x) = -2x + 3 g(x) = -2x - 4 We can see that k(x) = g(x) + 7, because -2x - 4 + 7 = -2x + 3. Therefore, you can obtain the graph of k(x) by shifting the graph of g(x) vertically upwards by 7 units. 3 done, 2 left today. You're making progress.