Recover the signs hidden by an absolute value · source reference

Printed question

The graphs of $y=f'(x)$ and $y=|f(x)|$ are shown below. The gradient graph has turning point $(-1,-2)$, intercepts $O$ and $(3,0)$, and asymptotes $y=0$, $x=2$. The absolute-value graph shows $(0,1)$, $(1,0)$, $(3,3)$ and asymptotes $y=0$, $x=2$.

(i) State the nature of all stationary points of the graph of $y=f(x)$. [2]

(ii) State the range of values of $x$ where (a) $f$ is decreasing, [2] (b) $f$ is concave upwards. [1]

(iii) Sketch the graph of $y=f(x)$ indicating clearly the asymptotes, coordinates of the stationary points and the intersections with the axes. [3]

[2021/DHS/Promo/7]

Printed working

(i) Maximum point $(3,-3)$, minimum point $(0,-1)$.

(ii)(a) Printed interval: $x\le0$ or $x\ge3$. (ii)(b) Printed interval: $-1\le x<2$.

(iii) Reconstructed $f$ is negative for $x<1$, crosses at $(1,0)$, positive for $1<x<2$, and negative for $x>2$, with extrema $(0,-1)$ and $(3,-3)$. Asymptotes are $y=0$ (to the left) and $x=2$.

Endpoint qualification

For strict derivative sign statements use $f'<0$ on $x<0$ or $x>3$, and $f''>0$ on $-1<x<2$. The printed closed endpoints express decreasing/convex interval conventions; derivative is zero at the displayed boundaries. The discontinuity at $2$ splits the increasing ranges.

Teacher notes

To sketch $y=f(x)$:

1. Curve $y=|f(x)|$ is obtained from $y=f(x)$ by reflecting the part where $y<0$ above the $x$-axis.

2. To obtain $y=f(x)$ reflect some parts of $y=|f(x)|$ below the $x$-axis.

3(a). From $y=f'(x)$, $x$-intercept $(0,0)$ becomes a stationary point at $x=0$ for $y=f(x)$.

3(b). $f'(x)>0$ for $0<x<3$, $x\ne2$: the gradient of $y=f(x)$ is positive, so $f$ is strictly increasing on the separate intervals $0<x<2$ and $2<x<3$. Reflect $|f(x)|$ where $0<x<1$ and $2<x<3$ below the axis.

3(c). $f'(x)<0$ for $x<0$ or $x>3$: the gradient of $y=f(x)$ is negative, so $f$ is strictly decreasing there. Reflect $|f(x)|$ where $x<0$ and $x>3$ below the axis.

Answer key

(i) Max $(3,-3)$, min $(0,-1)$. (ii)(a) $x\le0$ or $x\ge3$; (ii)(b) $-1\le x<2$. (iii) Same native reconstructed sketch.

Native diagrams