Question
Begin by graphing the absolute value function, f left parenthesis x right parenthesis equals StartAbsoluteValue x EndAbsoluteValuef(x)=x. Then use transformations of this graph to graph the given function.
h left parenthesis x right parenthesis equals StartAbsoluteValue x plus 1 EndAbsoluteValue plus 5h(x)=x+1+5
Question content area bottom
Part 1
What transformations are needed in order to obtain the graph of h(x) from the graph of f(x)? Select all that apply.
A.
Reflection about the y-axis
B.
Horizontal translation
C.
Vertical stretch/shrink
D.
Horizontal stretch/shrink
E.
Reflection about the x-axis
F.
Vertical translation
h left parenthesis x right parenthesis equals StartAbsoluteValue x plus 1 EndAbsoluteValue plus 5h(x)=x+1+5
Question content area bottom
Part 1
What transformations are needed in order to obtain the graph of h(x) from the graph of f(x)? Select all that apply.
A.
Reflection about the y-axis
B.
Horizontal translation
C.
Vertical stretch/shrink
D.
Horizontal stretch/shrink
E.
Reflection about the x-axis
F.
Vertical translation
Answers
GPT-4o mini
To graph the function \( h(x) = |x + 1| + 5 \) from the function \( f(x) = |x| \), we need to perform specific transformations. Let's analyze the transformations step by step:
1. **Starting Function:** The base function is \( f(x) = |x| \), which is a V-shaped graph that opens upwards and is centered at the origin (0,0).
2. **Transformations to Obtain \( h(x) \):**
- **Horizontal Translation:** The term \( x + 1 \) inside the absolute value indicates a horizontal translation to the left by 1 unit because when you add a positive number inside the function, it shifts left. This transforms \( |x| \) to \( |x + 1| \).
- **Vertical Translation:** The \( +5 \) outside of the absolute value indicates a vertical translation upwards by 5 units. After moving the graph, we lift it vertically. This transforms \( |x + 1| \) into \( |x + 1| + 5 \).
Given these transformations, the applicable options to obtain the graph of \( h(x) \) from \( f(x) \) are:
- **B. Horizontal translation (left by 1 unit)**
- **F. Vertical translation (up by 5 units)**
**Final Answer:**
Select B and F.
1. **Starting Function:** The base function is \( f(x) = |x| \), which is a V-shaped graph that opens upwards and is centered at the origin (0,0).
2. **Transformations to Obtain \( h(x) \):**
- **Horizontal Translation:** The term \( x + 1 \) inside the absolute value indicates a horizontal translation to the left by 1 unit because when you add a positive number inside the function, it shifts left. This transforms \( |x| \) to \( |x + 1| \).
- **Vertical Translation:** The \( +5 \) outside of the absolute value indicates a vertical translation upwards by 5 units. After moving the graph, we lift it vertically. This transforms \( |x + 1| \) into \( |x + 1| + 5 \).
Given these transformations, the applicable options to obtain the graph of \( h(x) \) from \( f(x) \) are:
- **B. Horizontal translation (left by 1 unit)**
- **F. Vertical translation (up by 5 units)**
**Final Answer:**
Select B and F.