1. What is the quotient in simplest form? State any restrictions on the variable.
z2−4z−3÷z+2z2+z−12
2. Describe the graph of y=12x−10−3
compared to the graph of y=1x
.
1. To simplify the quotient, first factor both the numerator and denominator:
z^2 - 4z - 3 = (z-3)(z+1)
z + 2z^2 + z - 12 = (z+4)(z-3)
So the expression simplifies to: (z-3)(z+1) / (z+4)(z-3)
To find the restrictions on the variable, we see that z cannot be equal to 3 or -4, as these values would result in division by zero.
2. The graph of y = 12x - 10 - 3 can be rewritten as y = 12x - 13. This is a linear equation with a slope of 12 and a y-intercept of -13. This means the graph is a straight line with a steep slope and will intersect the y-axis at -13.
On the other hand, the graph of y = x (or y = 1x) is also a straight line, but with a lower slope of 1 and a y-intercept of 0. This line will be less steep compared to the first line.
In summary, the graph of y = 12x - 13 will be steeper and shifted downwards compared to the graph of y = x.