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 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 [minus−10, 10] by [minus−40, 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. A downward U-shaped curve has a maximum on the positive x-axis and exits the viewing window in the third and fourth quadrants.
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 curve rises from left to right at a decreasing rate, passing through the negative x-axis. A second curve is to the right of the first curve and falls at an increasing rate through the positive x-axis.
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 in the third quadrant 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 [minus−10, 10] by [minus−40, 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. A downward U-shaped curve has a maximum on the positive x-axis and exits the viewing window in the third and fourth quadrants.
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 curve rises from left to right at a decreasing rate, passing through the negative x-axis. A second curve is to the right of the first curve and falls at an increasing rate through the positive x-axis.
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 in the third quadrant and passes through the positive and negative x-axis.
D.
Answers
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Answered by
GPT-4o mini
To find the vertex of the function \( y = 2x^2 + 4x - 28 \) and the appropriate graph, let's solve the problems step by step.
### Part 1: Finding the Vertex
The vertex of a quadratic function \( y = ax^2 + bx + c \) can be found using the formula:
\[
x = -\frac{b}{2a}
\]
Here, \( a = 2 \), \( b = 4 \), and \( c = -28 \).
Calculating the x-coordinate of the vertex:
\[
x = -\frac{4}{2 \cdot 2} = -\frac{4}{4} = -1
\]
Next, we need to find the y-coordinate by substituting \( x = -1 \) back into the function:
\[
y = 2(-1)^2 + 4(-1) - 28
\]
\[
y = 2(1) - 4 - 28
\]
\[
y = 2 - 4 - 28 = -30
\]
So the coordinates of the vertex are:
\[
(-1, -30)
\]
### Part 2: Choosing the Correct Graph
Now, let's analyze the possible graphs. The function \( y = 2x^2 + 4x - 28 \) is a parabola that opens upwards (since \( a = 2 > 0 \)) and has its vertex at \( (-1, -30) \).
We need to identify the graph that represents a parabola that opens upwards and that includes the vertex at \( (-1, -30) \).
- **Graph A** suggests a downward U-shaped curve, which does not represent our function.
- **Graph B** appears to depict two separate curves, which doesn't match our quadratic function either.
- **Graph C** represents an upward U-shaped curve. If its vertex lies in the third quadrant (which it does, since the vertex is at (-1,-30)), and since it passes through both the positive and negative x-axis, it seems to fit our quadratic.
- **Graph D** is not described, so we can't assess it.
Based on our analysis, the correct answer is:
- **Vertex**: \((-1, -30)\)
- **Graph**: **C** (the upward U-shaped curve with a minimum in the third quadrant).
### Part 1: Finding the Vertex
The vertex of a quadratic function \( y = ax^2 + bx + c \) can be found using the formula:
\[
x = -\frac{b}{2a}
\]
Here, \( a = 2 \), \( b = 4 \), and \( c = -28 \).
Calculating the x-coordinate of the vertex:
\[
x = -\frac{4}{2 \cdot 2} = -\frac{4}{4} = -1
\]
Next, we need to find the y-coordinate by substituting \( x = -1 \) back into the function:
\[
y = 2(-1)^2 + 4(-1) - 28
\]
\[
y = 2(1) - 4 - 28
\]
\[
y = 2 - 4 - 28 = -30
\]
So the coordinates of the vertex are:
\[
(-1, -30)
\]
### Part 2: Choosing the Correct Graph
Now, let's analyze the possible graphs. The function \( y = 2x^2 + 4x - 28 \) is a parabola that opens upwards (since \( a = 2 > 0 \)) and has its vertex at \( (-1, -30) \).
We need to identify the graph that represents a parabola that opens upwards and that includes the vertex at \( (-1, -30) \).
- **Graph A** suggests a downward U-shaped curve, which does not represent our function.
- **Graph B** appears to depict two separate curves, which doesn't match our quadratic function either.
- **Graph C** represents an upward U-shaped curve. If its vertex lies in the third quadrant (which it does, since the vertex is at (-1,-30)), and since it passes through both the positive and negative x-axis, it seems to fit our quadratic.
- **Graph D** is not described, so we can't assess it.
Based on our analysis, the correct answer is:
- **Vertex**: \((-1, -30)\)
- **Graph**: **C** (the upward U-shaped curve with a minimum in the third quadrant).
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