Let g(x) = x+c. The graph of f(x) = (a)/(x+p) + q has a vertical asymptote x=-p and a horizontal asymptote y=q. If the graph of g and the vertical asymptote of f intersect at the x-axis, and the graph

Right ~👻Masana, let's go. 6.1 The equation of graph g is g(x) = x+c. To find the x-intercept of g, we set g(x) = 0: x+c = 0 x = -c So the x-intercept of g is (-c; 0). The equation of graph f is f(x) = (a)/(x+p) + q. The vertical asymptote of f occurs when the denominator is zero: x+p = 0 x = -p The problem states that graph g and the vertical asymptote of f intersect at the x-axis. This means their x-intercepts are the same. Therefore, -c = -p, which implies c=p. Substituting c=p into the x-intercept of g: The coordinates of the x-intercept of g are (-p; 0). (-p; 0) 6.2 We need to determine the equation of f(x) = (a)/(x+p) + q. We already know c=p. Step 1: Use the information that g intersects the horizontal asymptote of f at x=1. The horizontal asymptote of f(x) is y=q. Since g(x) intersects y=q at x=1, we have g(1) = q. Substitute x=1 into g(x) = x+c: g(1) = 1+c So, q = 1+c. Since c=p, we have q = 1+p. Step 2: Use the information that f and g intersect on the y-axis. This means f(0) = g(0). Substitute x=0 into g(x) = x+c: g(0) = 0+c = c Substitute x=0 into f(x) = (a)/(x+p) + q: f(0) = (a)/(0+p) + q = (a)/(p) + q Equating f(0) and g(0): c = (a)/(p) + q Substitute q=1+c: c = (a)/(p) + (1+c) 0 = (a)/(p) + 1 (a)/(p) = -1 a = -p Step 3: Use the information that f and g intersect at x=3. This means f(3) = g(3). Substitute x=3 into g(x) = x+c: g(3) = 3+c Substitute x=3 into f(x) = (a)/(x+p) + q: f(3) = (a)/(3+p) + q Equating f(3) and g(3): 3+c = (a)/(3+p) + q Substitute a=-p and q=1+c: 3+c = (-p)/(3+p) + (1+c) Subtract (1+c) from both sides: (