Suppose \( (5,-2) \) is a point on the graph of \( y=g(x) \). (a) What point is on the graph of \( y=g(x+1)-6 \) ? (b) What point is on the graph of \( y=-5 g(x-5)+8 \) ? (c) What point is on the graph of \( y=g(5 x+20) \) ? (a) What point is on the graph of \( y=g(x+1)-6 \) ? (Type an ordered pair.)

If \( (5,-2) \) is a point on the graph of \( y=g(x) \), that means \( g(5) = -2 \). (a) For the transformation \( y = g(x+1) - 6 \), we first need to find the x-value that corresponds to 5. Since we are replacing \( x \) with \( x+1 \), we set \( x + 1 = 5 \), leading to \( x = 4 \). Now we plug this into the function: \( g(4 + 1) = g(5) = -2 \). Then we apply the -6 to get \( -2 - 6 = -8 \). Therefore, the point on the graph is \( (4, -8) \). (b) For the transformation \( y = -5 g(x-5) + 8 \), we need to adjust for the x-value again. Set \( x - 5 = 5 \) which gives \( x = 10 \). Now using this in the function: \( g(10 - 5) = g(5) = -2 \). We then compute the transformation: \( -5(-2) + 8 = 10 + 8 = 18 \). Thus, the point on the graph is \( (10, 18) \). (c) For \( y = g(5x + 20) \), to find the point corresponding to \( (5, -2) \), we need to solve \( 5x + 20 = 5 \). This means \( 5x = 5 - 20 \) leading to \( 5x = -15 \) or \( x = -3 \). At this value, we find: \( g(5(-3) + 20) = g(5) = -2 \). So the directed point becomes \( (-3, -2) \).