The cross-sectional profile of a custom-designed skate ramp is modeled by the curve C1C_1C1 with equation y=f(x)y = f(x)y=f(x), where f(x)=(6−x)(2x+5)2f(x) = (6 - x)(2x + 5)^2f(x)=(6−x)(2x+5)2 and xxx represents horizontal distance from a sensor in meters.
Sketch C1C_1C1 showing the coordinates of any point where the curve touches or crosses the coordinate axes.
Hence or otherwise, (i) find the values of xxx for which f(3x)=0f(3x) = 0f(3x)=0. (ii) find the value of the constant kkk such that the curve with equation y=f(x)+ky = f(x) + ky=f(x)+k passes through the origin.
A second curve C2C_2C2 has equation y=g(x)y = g(x)y=g(x), where g(x)=f(x−2)g(x) = f(x - 2)g(x)=f(x−2).
(c) (i) Find, in simplest form, g(x)g(x)g(x). You may leave your answer in a factorised form. (ii) Hence, or otherwise, find the yyy-intercept of curve C2C_2C2.
