Question
Which point should have an open circle in the graph of the piecewise function f(x)={−x−1 if x<1−x+2 if x≥1 ?(1 point)
Responses
(1,−2)
left parenthesis 1 comma negative 2 right parenthesis
(1,1)
left parenthesis 1 comma 1 right parenthesis
(−1,1)
left parenthesis negative 1 comma 1 right parenthesis
(−1,−2)
A company charges $15 per hour plus a $20 setup fee to rent a bouncy house for 3 hours or less. They charge $12 per hour and no setup fee if the bouncy house is rented for more than 3 hours. Sketch a graph of the piecewise function that models this scenario. Use the graph to select the correct equation for the function where x
is the number of hours and y
is the total cost.(1 point)
Responses
y={15+20x, 0<x≤312, x>3
y equals left brace line 1 15 plus 20 times x comma 0 less than x less than or equal to three line 2 12 comma x greater than 3
y={15+20, 0<x≤312, x>3
y equals left brace line 1 15 plus 20 comma 0 less than x less than or equal to three line 2 12 comma x greater than 3
y={15x+20, 0<x<312x, x≥3
y equals left brace line 1 15 times x + 20 comma 0 less than x less than three line 2 12 times x comma x greater than or equal to 3
y={15x+20, 0<x≤312x, x>3
Select the graph of the function y=(x−6)2−7.(1 point)
Responses
A. A U-shaped function is graphed on the coordinate plane.
Image with alt text: A U-shaped function is graphed on the coordinate plane.
B. A U-shaped function is graphed on the coordinate plane.
Image with alt text: A U-shaped function is graphed on the coordinate plane.
C A U-shaped function is graphed on the coordinate plane.
Image with alt text: A U-shaped function is graphed on the coordinate plane.
D U-shaped function is graphed on the coordinate plane.
Image with alt text: A U-shaped function is graphed on the coordinate plane.
Remaining Attempts : 3
Responses
(1,−2)
left parenthesis 1 comma negative 2 right parenthesis
(1,1)
left parenthesis 1 comma 1 right parenthesis
(−1,1)
left parenthesis negative 1 comma 1 right parenthesis
(−1,−2)
A company charges $15 per hour plus a $20 setup fee to rent a bouncy house for 3 hours or less. They charge $12 per hour and no setup fee if the bouncy house is rented for more than 3 hours. Sketch a graph of the piecewise function that models this scenario. Use the graph to select the correct equation for the function where x
is the number of hours and y
is the total cost.(1 point)
Responses
y={15+20x, 0<x≤312, x>3
y equals left brace line 1 15 plus 20 times x comma 0 less than x less than or equal to three line 2 12 comma x greater than 3
y={15+20, 0<x≤312, x>3
y equals left brace line 1 15 plus 20 comma 0 less than x less than or equal to three line 2 12 comma x greater than 3
y={15x+20, 0<x<312x, x≥3
y equals left brace line 1 15 times x + 20 comma 0 less than x less than three line 2 12 times x comma x greater than or equal to 3
y={15x+20, 0<x≤312x, x>3
Select the graph of the function y=(x−6)2−7.(1 point)
Responses
A. A U-shaped function is graphed on the coordinate plane.
Image with alt text: A U-shaped function is graphed on the coordinate plane.
B. A U-shaped function is graphed on the coordinate plane.
Image with alt text: A U-shaped function is graphed on the coordinate plane.
C A U-shaped function is graphed on the coordinate plane.
Image with alt text: A U-shaped function is graphed on the coordinate plane.
D U-shaped function is graphed on the coordinate plane.
Image with alt text: A U-shaped function is graphed on the coordinate plane.
Remaining Attempts : 3
Answers
GPT-4o mini
Let's tackle each part of your question step by step.
### 1. Open Circle in the Piecewise Function
The piecewise function you're given is:
\[
f(x) = \begin{cases}
-x - 1 & \text{if } x < 1 \\
-x + 2 & \text{if } x \geq 1
\end{cases}
\]
To determine which point should have an open circle, we need to evaluate \( f(1) \):
- For \( x < 1 \): \( f(1) \) is calculated using the first piece, but since 1 is not less than 1, we can't use this.
- For \( x \geq 1 \): \( f(1) = -1 + 2 = 1 \).
Thus, \( f(1) \) is defined and does not require an open circle on the point \( (1, 1) \). However, for \( x < 1 \), as \( x \) approaches 1, \( f(x) = -x - 1 \) approaches \( -2 \).
Therefore, the point that requires an open circle is \( (1, -2) \) on the first part, as it is not included in the function.
### Answer: **(1, −2)**
---
### 2. Cost Function for Bouncy House Rental
In this scenario, you are given the following conditions:
- For \( x \) (hours) less than or equal to 3: The cost is $15 per hour plus a $20 setup fee. So the total cost is:
\[
y = 15x + 20
\]
- For \( x \) greater than 3: The cost is $12 per hour with no setup fee. So the total cost is:
\[
y = 12x
\]
Now, we can express this as a piecewise function:
\[
y =
\begin{cases}
15x + 20 & \text{if } 0 < x \leq 3 \\
12x & \text{if } x > 3
\end{cases}
\]
### Correct Response: **y = {15x + 20, 0 < x ≤ 3; 12x, x > 3}**
---
### 3. Graph of the Function \( y = (x-6)^2 - 7 \)
The function \( y = (x-6)^2 - 7 \) represents a parabola that opens upward (since the coefficient of \((x-6)^2\) is positive).
- The vertex of this parabola is at \( (6, -7) \).
- The parabola will be symmetrical around the line \( x = 6 \).
Since the choices you have presented are vague without specific plots, generally, look for the graph that shows a U-shape opening upwards with the vertex at \( (6, -7) \).
### Unfortunately, I am unable to visually distinguish between images without joining context regarding their axes, shape, or vertex location.
If you have access to the specific graphs, choose the one that has the U-shape centered around \( x = 6 \) and \( y = -7 \).
---
Please, review your graph options based on the defined characteristics of each function. If you have further details or different options for the graphs, feel free to provide them!
### 1. Open Circle in the Piecewise Function
The piecewise function you're given is:
\[
f(x) = \begin{cases}
-x - 1 & \text{if } x < 1 \\
-x + 2 & \text{if } x \geq 1
\end{cases}
\]
To determine which point should have an open circle, we need to evaluate \( f(1) \):
- For \( x < 1 \): \( f(1) \) is calculated using the first piece, but since 1 is not less than 1, we can't use this.
- For \( x \geq 1 \): \( f(1) = -1 + 2 = 1 \).
Thus, \( f(1) \) is defined and does not require an open circle on the point \( (1, 1) \). However, for \( x < 1 \), as \( x \) approaches 1, \( f(x) = -x - 1 \) approaches \( -2 \).
Therefore, the point that requires an open circle is \( (1, -2) \) on the first part, as it is not included in the function.
### Answer: **(1, −2)**
---
### 2. Cost Function for Bouncy House Rental
In this scenario, you are given the following conditions:
- For \( x \) (hours) less than or equal to 3: The cost is $15 per hour plus a $20 setup fee. So the total cost is:
\[
y = 15x + 20
\]
- For \( x \) greater than 3: The cost is $12 per hour with no setup fee. So the total cost is:
\[
y = 12x
\]
Now, we can express this as a piecewise function:
\[
y =
\begin{cases}
15x + 20 & \text{if } 0 < x \leq 3 \\
12x & \text{if } x > 3
\end{cases}
\]
### Correct Response: **y = {15x + 20, 0 < x ≤ 3; 12x, x > 3}**
---
### 3. Graph of the Function \( y = (x-6)^2 - 7 \)
The function \( y = (x-6)^2 - 7 \) represents a parabola that opens upward (since the coefficient of \((x-6)^2\) is positive).
- The vertex of this parabola is at \( (6, -7) \).
- The parabola will be symmetrical around the line \( x = 6 \).
Since the choices you have presented are vague without specific plots, generally, look for the graph that shows a U-shape opening upwards with the vertex at \( (6, -7) \).
### Unfortunately, I am unable to visually distinguish between images without joining context regarding their axes, shape, or vertex location.
If you have access to the specific graphs, choose the one that has the U-shape centered around \( x = 6 \) and \( y = -7 \).
---
Please, review your graph options based on the defined characteristics of each function. If you have further details or different options for the graphs, feel free to provide them!