Given the graphs of f(x) = ax² + bx + c and g(x) = m/x-p + q, answer the following questions: Give the value(s) for which f is increasing? Determine the equation of f. Is g continuous or discontinuous? Give the equation of the asymptote(s) of g. Hence, determine the equation of g.

Here are the solutions to Question 3: 3.1.1 To draw the graphs of f(x) = x^2 - 2x - 8 and g(x) = 2^x + 2, you would need to plot the following key features: For f(x) = x^2 - 2x - 8 (Parabola): x-intercepts: Set f(x)=0 x^2 - 2x - 8 = 0 (x-4)(x+2) = 0. The x-intercepts are at x=4 and x=-2. y-intercept: Set x=0 f(0) = 0^2 - 2(0) - 8 = -8. The y-intercept is at y=-8. Turning point (vertex): x = -(b)/(2a) = -(-2)/(2(1)) = 1. f(1) = 1^2 - 2(1) - 8 = 1 - 2 - 8 = -9. The turning point is at (1, -9). The parabola opens upwards. For g(x) = 2^x + 2 (Exponential function): x-intercepts: Set g(x)=0 2^x + 2 = 0 2^x = -2. There are no x-intercepts. y-intercept: Set x=0 g(0) = 2^0 + 2 = 1 + 2 = 3. The y-intercept is at y=3. Asymptote: The horizontal asymptote is y=k for g(x) = a · b^x + k. Here, k=2, so the horizontal asymptote is y=2. 3.1.2 To write f(x) = x^2 - 2x - 8 in the form f(x) = a(x-p)^2 + q, we complete the square: f(x) = x^2 - 2x - 8 f(x) = (x^2 - 2x + 1) - 1 - 8 f(x) = (x-1)^2 - 9 The equation in the desired form is f(x) = (x-1)^2 - 9. 3.2.1 From the graph of f, the parabola opens upwards and its turning point is at x=1. The function f is increasing for all x-values to the right of the turning point. Therefore, f is increasing for x > 1. 3.2.2 From the graph, the x-intercepts of f are at x=-2 and x=4. We can write the equation in factored form as f(x) = a(x - (-2))(x - 4) = a(x+2)(x-4). The y-intercept is at (0, -8). Substitute this point into the equation: -8 = a(0+2)(0-4) -8 = a(2)(-4) -8 = -8a a = 1 So, the equation of f is f(x) = 1(x+2)(x-4). f(x) = x^2 - 4x + 2x - 8 f(x) = x^2 - 2x - 8 The equation of f is f(x) = x^2 - 2x - 8. 3.2.3 The graph of g is a hyperbola with a vertical asymptote at x=1. A function is discontinuous if there is a break or a jump in its graph. Since the graph of g has a vertical asymptote, it is not continuous at that point. Therefore, g is discontinuous. 3.2.4 From the graph of g: The vertical dashed line is at x=1. This is the vertical asymptote x=1. The horizontal dashed line is at y=-3. This is the horizontal asymptote y=-3. 3.2.5 The general form of the equation for g is g(x) = (m)/(x-p) + q. From the asymptotes identified in 3.2.4: The vertical asymptote is x=p, so p=1. The horizontal asymptote is y=q, so q=-3. Substitute these values into the general equation: g(x) = (m)/(x-1) - 3 From the graph, the y-intercept of g is at (0, -2). Substitute this point into the equation to find m: -2 = (m)/(0-1) - 3 -2 = (m)/(-1) - 3 -2 = -m - 3 Add 3 to both sides: 1 = -m m = -1 Substitute m=-1 back into the equation: g(x) = (-1)/(x-1) - 3 The equation of g is g(x) = (-1)/(x-1) - 3. Last free one today — make it count tomorrow, or type /upgrade for unlimited.