Question
What is the standard form equation of the graph below?
graph shows a line passing through -5 and -3 and a curve at -4,2
(1 point)
Responses
y=−2x2−16x−30
y is equal to negative 2 x squared minus 16 x minus 30
y=4x2−10x+5
y is equal to 4 x squared minus 10 x plus 5
y=x2−6x+1
y is equal to x squared minus 6 x plus 1
y=−4x2+10x+15
graph shows a line passing through -5 and -3 and a curve at -4,2
(1 point)
Responses
y=−2x2−16x−30
y is equal to negative 2 x squared minus 16 x minus 30
y=4x2−10x+5
y is equal to 4 x squared minus 10 x plus 5
y=x2−6x+1
y is equal to x squared minus 6 x plus 1
y=−4x2+10x+15
Answers
GPT-4o mini
To find the standard form equation of the graph you described, we first need to identify whether the graph represents a linear function or a quadratic function since you mentioned both a line and a curve.
The graph has a curve at the point (-4, 2) and it passes through the points (-5, y) and (-3, y) for the line. Since the options given are quadratic equations, we will assume we are tasked with finding a quadratic equation that meets certain conditions.
To build a quadratic equation in standard form \( y = ax^2 + bx + c \), we will use the points you provided. The points where the curve has critical information are (-4, 2), which likely represents a vertex or a known point on the curve.
Given that none of the information explicitly identifies points on the line or extra points for certainty, we'll reason primarily from the point (-4, 2).
### Quadratic Vertex Form Approach
To derive a quadratic function based on the vertex form, we can start from the standard vertex format:
\[
y = a(x-h)^2 + k
\]
where (h, k) is the vertex.
Here, h = -4 and k = 2:
\[
y = a(x + 4)^2 + 2
\]
Next, we need to determine a using additional data points or the behavior of the curve.
To determine a value for \( a \), we'll check the values of the possible answers when x takes on values between -5 and -3:
1. Checking \( y = -2x^2 - 16x - 30 \):
- At \( x = -4 \):
\[
y = -2(-4)^2 - 16(-4) - 30 = -2(16) + 64 - 30 = -32 + 64 - 30 = 2
\]
2. Checking \( y = 4x^2 - 10x + 5 \):
- At \( x = -4 \):
\[
y = 4(-4)^2 - 10(-4) + 5 = 4(16) + 40 + 5 = 64 + 40 + 5 = 109
\]
3. Checking \( y = x^2 - 6x + 1 \):
- At \( x = -4 \):
\[
y = (-4)^2 - 6(-4) + 1 = 16 + 24 + 1 = 41
\]
4. Checking \( y = -4x^2 + 10x + 15 \):
- At \( x = -4 \):
\[
y = -4(-4)^2 + 10(-4) + 15 = -4(16) - 40 + 15 = -64 - 40 + 15 = -89
\]
From this evaluation, only the first equation \( y = -2x^2 - 16x - 30 \) equates to 2 when \( x = -4 \).
Therefore, the answer is:
\[
\boxed{y = -2x^2 - 16x - 30}
\]
The graph has a curve at the point (-4, 2) and it passes through the points (-5, y) and (-3, y) for the line. Since the options given are quadratic equations, we will assume we are tasked with finding a quadratic equation that meets certain conditions.
To build a quadratic equation in standard form \( y = ax^2 + bx + c \), we will use the points you provided. The points where the curve has critical information are (-4, 2), which likely represents a vertex or a known point on the curve.
Given that none of the information explicitly identifies points on the line or extra points for certainty, we'll reason primarily from the point (-4, 2).
### Quadratic Vertex Form Approach
To derive a quadratic function based on the vertex form, we can start from the standard vertex format:
\[
y = a(x-h)^2 + k
\]
where (h, k) is the vertex.
Here, h = -4 and k = 2:
\[
y = a(x + 4)^2 + 2
\]
Next, we need to determine a using additional data points or the behavior of the curve.
To determine a value for \( a \), we'll check the values of the possible answers when x takes on values between -5 and -3:
1. Checking \( y = -2x^2 - 16x - 30 \):
- At \( x = -4 \):
\[
y = -2(-4)^2 - 16(-4) - 30 = -2(16) + 64 - 30 = -32 + 64 - 30 = 2
\]
2. Checking \( y = 4x^2 - 10x + 5 \):
- At \( x = -4 \):
\[
y = 4(-4)^2 - 10(-4) + 5 = 4(16) + 40 + 5 = 64 + 40 + 5 = 109
\]
3. Checking \( y = x^2 - 6x + 1 \):
- At \( x = -4 \):
\[
y = (-4)^2 - 6(-4) + 1 = 16 + 24 + 1 = 41
\]
4. Checking \( y = -4x^2 + 10x + 15 \):
- At \( x = -4 \):
\[
y = -4(-4)^2 + 10(-4) + 15 = -4(16) - 40 + 15 = -64 - 40 + 15 = -89
\]
From this evaluation, only the first equation \( y = -2x^2 - 16x - 30 \) equates to 2 when \( x = -4 \).
Therefore, the answer is:
\[
\boxed{y = -2x^2 - 16x - 30}
\]