Estimate the value of $D$

Question

Q: In the figure below the graph of $f(x)$ is given in solid blue line; the graph of $g(x)=Af(Bx+C)+D$ is in dotted red line.

Estimate the value of $D$. my try: I know this, $$Q(x, y)\longrightarrow Q'\left(\frac{x-C}{B}, A.y+D\right)$$ so $$A(-3, -4)\longrightarrow A'(-12, -1)$$ $$B(10, 1)\longrightarrow B'(1, 14)$$ $$-4A+D=-1, \ \ A+D=14$$ solve the system of equations $$(A,D)=(3, 11)$$ but the answer is $D=2.$

Any help would be appreciated.

Answer 1

First we notice that $A$ is negative since the function is flipped about the $y-$ axis before being shifted further. We notice that because in $f(x)$ the left horizontal is the absolute minimum while in $g(x)$ the right horizontal is the absolute maximum.

Next we look at the constant parts (horizontal) of the functions to find $A$ and $D$. We concentrate on the areas where $f(x)$ and $g(x)$ are relatively constant to avoid having to deal with $B$ and $C$

We notice that $f(x)=-4$ corresponds with $g(x)=14$ and $f(x)=1$ corresponds with $g(x)=-1$. This leads us to the system of equation $$\begin{cases} 14=A(-4)+D\\ -1=A(1)+D \end{cases}$$

With a solution of $A=-3$ and $D=2$

Answer 2

HINT: Start by looking at the asymptotic behaviour $x\to\pm\infty$ of both functions $f$ and $g$. What can you deduce about the unknown coefficients? Then look at the zeros of $f$.