Final answer:
To find the values of a and b for a curve tangent to the line y = x at the origin, more information about the curve is needed. The slope at the tangent point is 1, but a and b remain undetermined without a specific equation for the curve.
Step-by-step explanation:
The student is looking to find the values of a and b for a curve that passes through the point (1,8) and is tangent to the line represented by y = ax + b at the origin. The information provided about a line with the equation y = 9 + 3x helps us understand the meaning of the slope and y-intercept in the context of a straight line. However, it seems we do not have all the specific details needed for the curve in question, such as its equation or other defining features.
Since the curve is assumed to be tangent to the line y = x at the origin, we know that at the origin the slope of the curve will also be 1 (as the slope of y = x is 1). However, without additional information about the curve itself, we can't solve for a and b.
For constructing tables and graphs, we plug in values for x, calculate the corresponding y values, and plot them on a graph. The slope of the line corresponds to the increase in y for every unit increase in x and the y-intercept is the point where the line crosses the y-axis.