Question

1. the function k is constructed by applying three - transformations to the graph of f in this order: a horizontal dilation by a factor of 4, a vertical dilation by a factor of 1/2, and a vertical translation by 3 units. if k(x)=aa(bx)+c, find the values of a, b and c.
2. the graph of y = f(x), consisting of two line segments and a semi - circle, is shown for - 4≤x≤3. sketch a graph of g on the same axes above where g(x)=f(x - 2).

the graph of y = f(x), consisting of two line segments and a semi - circle, is shown for - 4≤x≤3. sketch a graph of h on the same axes above where h(x)=2f(x + 1).

Explanation:

Step1: Recall horizontal - shift rule

For a function $y = f(x - c)$, the graph of $y = f(x)$ is shifted $c$ units to the right. Given $g(x)=f(x - 2)$, the graph of $f(x)$ will be shifted 2 units to the right.

Step2: Identify key - points of $f(x)$

Let's assume some key - points on $y = f(x)$. For example, if $(x_1,y_1)$ is a point on $y = f(x)$, then the corresponding point on $y = g(x)$ is $(x_1 + 2,y_1)$.

Step3: Sketch the graph of $g(x)$

Take each key - point of $f(x)$ (such as endpoints of line segments and critical points of the semi - circle), add 2 to the $x$ - coordinate, and plot these new points. Then connect the points in the same way as in the graph of $f(x)$ to get the graph of $g(x)$.

For the function $h(x)=2f(x + 1)$:

Step1: Recall horizontal - shift and vertical - dilation rules

The $x+1$ inside the function $f$ causes a horizontal shift of the graph of $f(x)$ 1 unit to the left. The coefficient 2 in front of $f$ causes a vertical dilation by a factor of 2.

Step2: Identify key - points of $f(x)$

Let $(x_0,y_0)$ be a point on $y = f(x)$. For the horizontal shift, the $x$ - coordinate of the new point for the shifted function is $x_0-1$. For the vertical dilation, the $y$ - coordinate of the final point on $y = h(x)$ is $2y_0$. So the point on $y = h(x)$ corresponding to $(x_0,y_0)$ on $y = f(x)$ is $(x_0 - 1,2y_0)$.

Step3: Sketch the graph of $h(x)$

Take each key - point of $f(x)$ (endpoints of line segments, critical points of the semi - circle), subtract 1 from the $x$ - coordinate and multiply the $y$ - coordinate by 2. Plot these new points and connect them to get the graph of $h(x)$.

Answer:

To sketch $g(x)=f(x - 2)$, shift the graph of $f(x)$ 2 units to the right. To sketch $h(x)=2f(x + 1)$, shift the graph of $f(x)$ 1 unit to the left and then vertically dilate it by a factor of 2.