Question

monitoring progress help in english and spanish at write a function g whose graph represents the indicated transformation of the graph of f. use a graphing calculator to check your answer. 1. f(x) = 3x; translation 5 units up 2. f(x) = |x| - 3; translation 4 units to the right 3. f(x) = -|x + 2| - 1; reflection in the x - axis 4. f(x) = \frac{1}{2}x + 1; reflection in the y - axis

Explanation:

Let's solve each sub - question one by one:

Sub - question 1: $f(x)=3x$; translation 5 units up

When we translate a function $y = f(x)$ $k$ units up, the new function $g(x)$ is given by $g(x)=f(x)+k$. Here, $f(x) = 3x$ and $k = 5$.

Step 1: Recall the vertical translation rule

The rule for vertical translation of a function $y = f(x)$ by $k$ units up is $g(x)=f(x)+k$.

Step 2: Apply the rule to $f(x)=3x$ with $k = 5$

Substitute $f(x)=3x$ and $k = 5$ into the formula: $g(x)=3x + 5$.

Sub - question 2: $f(x)=\vert x\vert-3$; translation 4 units to the right

When we translate a function $y = f(x)$ $h$ units to the right, the new function $g(x)$ is given by $g(x)=f(x - h)$. Here, $f(x)=\vert x\vert-3$ and $h = 4$.

Step 1: Recall the horizontal translation rule

The rule for horizontal translation of a function $y = f(x)$ by $h$ units to the right is $g(x)=f(x - h)$.

Step 2: Apply the rule to $f(x)=\vert x\vert-3$ with $h = 4$

Substitute $x$ with $x - 4$ in $f(x)$: $g(x)=\vert x - 4\vert-3$.

Sub - question 3: $f(x)=-\vert x + 2\vert-1$; reflection in the $x$ - axis

When we reflect a function $y = f(x)$ in the $x$ - axis, the new function $g(x)$ is given by $g(x)=-f(x)$.

Step 1: Recall the reflection rule over the $x$ - axis

The rule for reflecting a function $y = f(x)$ over the $x$ - axis is $g(x)=-f(x)$.

Step 2: Apply the rule to $f(x)=-\vert x + 2\vert-1$

$g(x)=-(-\vert x + 2\vert-1)=\vert x + 2\vert+1$.

Sub - question 4: $f(x)=\frac{1}{2}x + 1$; reflection in the $y$ - axis

Answer:

s:

1. $g(x)=3x + 5$
2. $g(x)=\vert x - 4\vert-3$
3. $g(x)=\vert x + 2\vert+1$
4. $g(x)=-\frac{1}{2}x + 1$