Duplicate Question
The question on this page has been marked as a duplicate question.
Original Question
Consider the function y equals 2 x squared plus 4 x minus 28y=2x2+4x−28. (a) Give the coordinates of the vertex of the graph o...Question
Consider the function y equals 3 x squared plus 12 x minus 23.
(a) Give the coordinates of the vertex of the graph of the function.
(b) Graph the function on a window that includes the vertex.
Question content area bottom
Part 1
(a) The vertex of the graph of the function is at the point
enter your response here. (Type an ordered pair.)
Part 2
(b) Choose the correct graph below. Each graph shown below has a window of [minus10, 10] by [minus40, 10].
A.
A coordinate system has a horizontal x-axis from negative 10 to 10 in increments of 1 and a vertical y-axis from negative 40 to 10 in increments of 5. An upward U-shaped curve has a minimum in the third quadrant and passes through the positive and negative x-axis.
B.
A coordinate system has a horizontal x-axis from negative 10 to 10 in increments of 1 and a vertical y-axis from negative 40 to 10 in increments of 5. A downward U-shaped curve has a maximum on the positive x-axis and exits the viewing window in the third and fourth quadrants.
C.
A coordinate system has a horizontal x-axis from negative 10 to 10 in increments of 1 and a vertical y-axis from negative 40 to 10 in increments of 5. An upward U-shaped curve has a minimum on the negative y-axis and passes through the positive and negative x-axis.
D.
(a) Give the coordinates of the vertex of the graph of the function.
(b) Graph the function on a window that includes the vertex.
Question content area bottom
Part 1
(a) The vertex of the graph of the function is at the point
enter your response here. (Type an ordered pair.)
Part 2
(b) Choose the correct graph below. Each graph shown below has a window of [minus10, 10] by [minus40, 10].
A.
A coordinate system has a horizontal x-axis from negative 10 to 10 in increments of 1 and a vertical y-axis from negative 40 to 10 in increments of 5. An upward U-shaped curve has a minimum in the third quadrant and passes through the positive and negative x-axis.
B.
A coordinate system has a horizontal x-axis from negative 10 to 10 in increments of 1 and a vertical y-axis from negative 40 to 10 in increments of 5. A downward U-shaped curve has a maximum on the positive x-axis and exits the viewing window in the third and fourth quadrants.
C.
A coordinate system has a horizontal x-axis from negative 10 to 10 in increments of 1 and a vertical y-axis from negative 40 to 10 in increments of 5. An upward U-shaped curve has a minimum on the negative y-axis and passes through the positive and negative x-axis.
D.
Answers
Answered by
GPT-4o mini
To analyze the function \( y = 3x^2 + 12x - 23 \), we can find the vertex and graph it.
### Part 1: Finding the Vertex
The vertex of a quadratic function represented in the form \( y = ax^2 + bx + c \) can be found using the formula:
\[
x = -\frac{b}{2a}
\]
For the given function:
- \( a = 3 \)
- \( b = 12 \)
Plugging in the values:
\[
x = -\frac{12}{2(3)} = -\frac{12}{6} = -2
\]
Now, we can substitute \( x = -2 \) back into the function to find the corresponding \( y \)-coordinate:
\[
y = 3(-2)^2 + 12(-2) - 23
\]
\[
y = 3(4) - 24 - 23
\]
\[
y = 12 - 24 - 23 = 12 - 47 = -35
\]
Thus, the vertex of the graph of the function is at the point \((-2, -35)\).
### Part 2: Select the Correct Graph
Since the function has a positive coefficient for \( x^2 \) (\( a = 3 > 0 \)), it will be a **U-shaped** parabola, which means the vertex is a **minimum** point.
Given the coordinates of the vertex \((-2, -35)\), we check the description of the graph:
- **Graph A**: Upward U-shaped curve, minimum in the third quadrant. (This fits our vertex being at \((-2, -35)\).)
- **Graph B**: Downward U-shaped curve. (Does not fit.)
- **Graph C**: Upward U-shaped curve, minimum on the negative y-axis. (This does not specify if it matches the vertex location.)
- **Graph D**: (No description provided.)
Based on the details, **Graph A** is most likely the correct graph since it matches the characteristics of the quadratic function and the vertex location.
### Final Answers
**Part 1:**
The vertex of the graph of the function is at the point \( (-2, -35) \).
**Part 2:**
The correct graph is **A**.
### Part 1: Finding the Vertex
The vertex of a quadratic function represented in the form \( y = ax^2 + bx + c \) can be found using the formula:
\[
x = -\frac{b}{2a}
\]
For the given function:
- \( a = 3 \)
- \( b = 12 \)
Plugging in the values:
\[
x = -\frac{12}{2(3)} = -\frac{12}{6} = -2
\]
Now, we can substitute \( x = -2 \) back into the function to find the corresponding \( y \)-coordinate:
\[
y = 3(-2)^2 + 12(-2) - 23
\]
\[
y = 3(4) - 24 - 23
\]
\[
y = 12 - 24 - 23 = 12 - 47 = -35
\]
Thus, the vertex of the graph of the function is at the point \((-2, -35)\).
### Part 2: Select the Correct Graph
Since the function has a positive coefficient for \( x^2 \) (\( a = 3 > 0 \)), it will be a **U-shaped** parabola, which means the vertex is a **minimum** point.
Given the coordinates of the vertex \((-2, -35)\), we check the description of the graph:
- **Graph A**: Upward U-shaped curve, minimum in the third quadrant. (This fits our vertex being at \((-2, -35)\).)
- **Graph B**: Downward U-shaped curve. (Does not fit.)
- **Graph C**: Upward U-shaped curve, minimum on the negative y-axis. (This does not specify if it matches the vertex location.)
- **Graph D**: (No description provided.)
Based on the details, **Graph A** is most likely the correct graph since it matches the characteristics of the quadratic function and the vertex location.
### Final Answers
**Part 1:**
The vertex of the graph of the function is at the point \( (-2, -35) \).
**Part 2:**
The correct graph is **A**.
There are no human answers yet. A form for humans to post answers is coming very soon!