Question

a basic function is given along with a new function f that can be obtained by a series of transformations of the given basic function. list the order of transformations required to sketch the graph of the new function. basic function: y = x^2; new function: f(x)= -((x + 10)^2)+2 choose the correct answer. a. 1. horizontal shift to the left 10 units; 2. reflection about the y - axis; 3. vertical stretch by a factor of 1; 4. vertical shift upward 2 units. b. 1. vertical shift upward 10 units; 2. vertical compression by a factor of 2; 3. horizontal shift to the left 2 units; 4. horizontal stretch by a factor of 1/4. c. 1. vertical stretch by a factor of ; 2. horizontal shift to the left 10 units; 3. vertical shift upward 2 units. d. 1. reflection about the y - axis; 2. vertical compression by a factor of 1/4; 3. vertical shift upward 2 units. e. 1. horizontal shift to the right 10 units; 2. reflection about the x - axis; 3. vertical shift downward 2 units.

Explanation:

Step1: Analyze horizontal shift

For the basic function $y = x^{2}$, in the new function $f(x)=-(x + 10)^{2}+2$, the $x$ is replaced with $x + 10$. According to the rule of horizontal - shift $y = f(x + h)$ shifts the graph of $y = f(x)$ to the left by $h$ units. Here $h = 10$, so there is a horizontal shift to the left 10 units.

Step2: Analyze reflection

The negative sign in front of $(x + 10)^{2}$ in $f(x)=-(x + 10)^{2}+2$ reflects the graph of $y=(x + 10)^{2}$ about the $x$-axis. In the options, the correct reflection description is not given exactly as such, but we note that the negative sign is for reflection about the $x$-axis.

Step3: Analyze vertical shift

The + 2 at the end of $f(x)=-(x + 10)^{2}+2$ shifts the graph of $y=-(x + 10)^{2}$ upward by 2 units.

The correct order of transformations is: horizontal shift to the left 10 units, reflection about the $x$-axis (not $y$-axis as some options wrongly state), and vertical shift upward 2 units. Among the given options, the closest correct one is:

1. Horizontal shift to the left 10 units; 2. Reflection about the $y$-axis (wrongly stated in the option but we know it should be $x$-axis); 3. Vertical shift upward 2 units.

Answer:

A. 1. Horizontal shift to the left 10 units; 2. Reflection about the y - axis; 3. Vertical shift upward 2 units (Note: the reflection should be about the x - axis in reality but this option is the closest in terms of the other correct shifts)