A curve $y=\mathrm{h}\left( x \right)$ intersects the $x$-axis at point $A$ with coordinates \

Question

A curve $y=\mathrm{h}\left( x \right)$ intersects the $x$-axis at point $A$ with coordinates $\left( a,0 \right)$, where $a\ne 0$, and has a horizontal asymptote with equation $y=b$, where $b$ is a non-zero constant. For each of the following curves, state, if it is possible to do so, the equation of the horizontal asymptote and the coordinates of the point where the curve intersects the $x$-axis. $y=2\mathrm{h}\left( x+1 \right)$ $y=\mathrm{h}\left( -x \right)$ $y=\frac{1}{\mathrm{h}\left( x \right)}$[5] It is given that ${\mathrm{f}(x) = \begin{cases} \sqrt{x} & \text{for}\hspace{0.5em} 0 \le x \le 1, \ -\frac{1}{2}x + \frac{3}{2} & \text{for}\hspace{0.5em} 1 < x \le 3, \end{cases}}$ and that $\mathrm{f}\left( x \right)=\mathrm{f}\left( x+3 \right)$ for all real values of $x$. Sketch the graph of $y=\mathrm{f}\left( x \right)$ for $-3\le x\le 5$.[3] Hence find the value of $\displaystyle\displaystyle \displaystyle\int_{-3}^{5}{\mathrm{f}\left( x \right)\mathrm{d}x}$.[2]

Accepted Answer

(a)(i) ${y=2b\;; (a-1,0)}$ (ii) ${y=b\;; (-a,0)}$ (iii) ${y=\frac{1}{b}\;; 	ext{inconclusive}}$ (b)(ii) ${4.75}$

2023 MI P1 Q9

2023 MI P1 Q9

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