Question

1. write a formula for $f(x)$ that satisfies the following

(a) the graph is obtained by shifting the graph of $f(x)=\sqrt{x}$ up 4 units and then reflecting on the $x - axis$
(b) the graph is obtained by shifting the graph of $f(x)=x^{2}$ up 2 units, 3 units left and then reflecting on the $x - axis$

1. plot the graphs obtained in question 1.
2. find the domain of the graphs obtained in question 1.
3. write a function for $t(x)$ which is obtained by stretching vertically $f(x)$ by a factor of 4 and reflecting $f(x)$ across the $y - axis$

Explanation:

Step1: For part (a) - vertical shift

The original function is $f(x)=\sqrt{x}$. Shifting it up 4 units gives $y = \sqrt{x}+4$.

Step2: For part (a) - reflection on x - axis

Reflecting $y=\sqrt{x}+4$ on the $x -$axis changes the sign of the function, so $f(x)=-(\sqrt{x}+4)=-\sqrt{x}-4$.

Step3: For part (b) - horizontal and vertical shifts

The original function is $f(x)=x^{2}$. Shifting it up 2 units gives $y = x^{2}+2$, and then shifting 3 units left gives $y=(x + 3)^{2}+2$.

Step4: For part (b) - reflection on x - axis

Reflecting $y=(x + 3)^{2}+2$ on the $x -$axis changes the sign of the function, so $f(x)=-((x + 3)^{2}+2)=-(x + 3)^{2}-2$.

Step5: For part 4 - vertical stretch

Let the original function be $f(x)$. Stretching it vertically by a factor of 4 gives $y = 4f(x)$.

Step6: For part 4 - reflection on y - axis

Reflecting $y = 4f(x)$ across the $y -$axis replaces $x$ with $-x$, so $T(x)=4f(-x)$.

Answer:

1. (a) $f(x)=-\sqrt{x}-4$

(b) $f(x)=-(x + 3)^{2}-2$

1. $T(x)=4f(-x)$

(Note: Plotting the graphs (part 2) and finding the domain (part 3) are not fully completed in this response as the main focus was on the formula - writing part. For the domain of $y =-\sqrt{x}-4$, the domain is $x\geq0$ since the square - root function $\sqrt{x}$ requires $x\geq0$. For $y=-(x + 3)^{2}-2$, the domain is all real numbers, i.e., $x\in(-\infty,\infty)$ as it is a polynomial function. For graph - plotting, you can use a table of values for each function to plot the points on the given grids.)