Practice section 9 of 9

The given graph $y=f(x)$ has asymptotes $x=-1$, $y=-2$, turning point $A=(1,3)$, and zeros $O=(0,0)$, $B=(-2,0)$, $C=(4,0)$.

(a) State the turning point of $y=1+\frac13f(x)$. [1]

(b) Sketch $y=|f(x)|$, labelling asymptotes and the points corresponding to A,B,C. [2]

(c) Sketch $y=-f(2x)$ with those labels. [3]

(d) Sketch $y=1/f(x)$, labelling asymptotes, turning points and axis crossings. [3]

A reflects in the y-axis; B translates 4 units in the positive x-direction; C scales parallel to y by 2. A curve undergoes A,B,C in that order, giving $y^2=-4x$. Determine its equation before the transformations. [3]

A translates 2 units in the negative x-direction; B reflects in the x-axis; C scales parallel to y by 2 with the x-axis invariant. Applied in order A,B,C, the resulting equation is $y=-2(x+3)/(x+2)$. Find the equation before the transformations. [3]

(a) The supplied graph $y=f(x)$ has asymptotes y=0, y=2, x=−1, x=1, maximum (3,3), minimum at O and the right zero (2,0) shown. Sketch:

(i) $y=f(|1+x|)$; [3]

(ii) $y=1/f(x)$. [3]

(b) Alfred and Betty are to transform $y=f(x)$ to $y=5f(2x+1)-3$, with original $f(x)=x^2$ and target $y=5(2x+1)^2-3$.

Alfred’s steps, in order: 1.Scale parallel to y by 5; 2.Translate down 3; 3. Scale parallel to x by 1/2; 4. Translate along x by −1.

Betty’s steps, in order: 1.Translate along x by −1; 2. Scale parallel to x by 1/2; 3. Scale parallel to y by 5; 4. Translate down 3.

One person made an error in one step.

(i) Identify the person and incorrect step. [2]

(ii) Give the final equation resulting from that incorrect sequence. [1]

The supplied graphs are $y=f(|x|)$ and $y=|f(x)|$. Both have maximum (4, a) and vertical tangents x=0, x=6. The first is even, with maxima (−4, a), (4, a), zeros (−6,0), (0,0), (6,0), and slant asymptotes y=x+2 on the left and y=−x+2 on the right. The second is nonnegative, with zeros (0,0), (6,0), and slant asymptotes y=−x+2 on the left and y=x−2 on the right.

Given f itself has only one oblique asymptote, sketch $y=f(x)$, stating asymptotes, turning points and axial intercepts wherever possible. [2]