Question
The graph shows a system consisting of a linear equation and a quadratic equation.
What is the solution to the system?
Responses
(5, 12)
begin ordered pair 5 comma 12 end ordered pair
(3, 4) and (5, 12)
begin ordered pair 3 comma 4 end ordered pair and begin ordered pair 5 comma 12 end ordered pair
3 and 5
3 and 5
4 and 12
4 and 12
(3, 4)
begin ordered pair 3 comma 4 end ordered pair
A line and a parabola are graphed on a coordinate plane. The horizontal x-axis ranges from negative 1 to 6 in increments of 1. The vertical y-axis ranges from negative 1 to 14 in increments of 1. The vertex of the parabola is located at begin ordered pair 2 comma 3 end ordered pair. The parabola opens upward. It passes through begin ordered pair 0 comma 7 end ordered pair and begin ordered pair 4 comma 7 end ordered pair. A straight line passes through begin ordered pair 2 comma 0 end ordered pair and begin ordered pair 5 comma 12 end ordered pair.
What is the solution to the system?
Responses
(5, 12)
begin ordered pair 5 comma 12 end ordered pair
(3, 4) and (5, 12)
begin ordered pair 3 comma 4 end ordered pair and begin ordered pair 5 comma 12 end ordered pair
3 and 5
3 and 5
4 and 12
4 and 12
(3, 4)
begin ordered pair 3 comma 4 end ordered pair
A line and a parabola are graphed on a coordinate plane. The horizontal x-axis ranges from negative 1 to 6 in increments of 1. The vertical y-axis ranges from negative 1 to 14 in increments of 1. The vertex of the parabola is located at begin ordered pair 2 comma 3 end ordered pair. The parabola opens upward. It passes through begin ordered pair 0 comma 7 end ordered pair and begin ordered pair 4 comma 7 end ordered pair. A straight line passes through begin ordered pair 2 comma 0 end ordered pair and begin ordered pair 5 comma 12 end ordered pair.
Answers
Answered by
GPT-4o mini
To find the solution to the system of a linear equation and a quadratic equation represented in your description, we need to identify the points where the line and the parabola intersect.
From your description:
- The quadratic has a vertex at (2, 3) and opens upward.
- The points (0, 7) and (4, 7) lie on the parabola, indicating a certain symmetry.
- The line passes through points (2, 0) and (5, 12).
### Finding the Linear Equation:
The slope \( m \) of the line can be calculated using the two points:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{12 - 0}{5 - 2} = \frac{12}{3} = 4.
\]
Using point-slope form, \( y - y_1 = m(x - x_1) \) with point (2, 0):
\[
y - 0 = 4(x - 2) \implies y = 4x - 8.
\]
### Finding the Quadratic Equation:
Given the vertex (2, 3) and using the vertex form \( y = a(x - 2)^2 + 3 \), we can determine \( a \) by using one of the other points on the parabola, say (0, 7):
\[
7 = a(0 - 2)^2 + 3 \implies 7 = 4a + 3 \implies 4 = 4a \implies a = 1.
\]
So the quadratic equation is:
\[
y = (x - 2)^2 + 3.
\]
### Finding Points of Intersection:
To find the points of intersection between the line and the parabola, we set the equations equal to each other:
\[
4x - 8 = (x - 2)^2 + 3.
\]
Expanding the quadratic:
\[
4x - 8 = x^2 - 4x + 4 + 3.
\]
Rearranging gives:
\[
0 = x^2 - 8x + 15.
\]
Factoring the quadratic:
\[
0 = (x - 3)(x - 5).
\]
This gives us:
\[
x = 3 \quad \text{and} \quad x = 5.
\]
### Finding Corresponding y-values:
1. For \( x = 3 \):
\[
y = 4(3) - 8 = 12 - 8 = 4 \quad \Rightarrow \quad (3, 4).
\]
2. For \( x = 5 \):
\[
y = 4(5) - 8 = 20 - 8 = 12 \quad \Rightarrow \quad (5, 12).
\]
### Conclusion:
The solutions to the system are the points of intersection, which are:
- \( (3, 4) \)
- \( (5, 12) \)
Thus, the correct response for the solutions to the system is:
**(3, 4) and (5, 12)**.
From your description:
- The quadratic has a vertex at (2, 3) and opens upward.
- The points (0, 7) and (4, 7) lie on the parabola, indicating a certain symmetry.
- The line passes through points (2, 0) and (5, 12).
### Finding the Linear Equation:
The slope \( m \) of the line can be calculated using the two points:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{12 - 0}{5 - 2} = \frac{12}{3} = 4.
\]
Using point-slope form, \( y - y_1 = m(x - x_1) \) with point (2, 0):
\[
y - 0 = 4(x - 2) \implies y = 4x - 8.
\]
### Finding the Quadratic Equation:
Given the vertex (2, 3) and using the vertex form \( y = a(x - 2)^2 + 3 \), we can determine \( a \) by using one of the other points on the parabola, say (0, 7):
\[
7 = a(0 - 2)^2 + 3 \implies 7 = 4a + 3 \implies 4 = 4a \implies a = 1.
\]
So the quadratic equation is:
\[
y = (x - 2)^2 + 3.
\]
### Finding Points of Intersection:
To find the points of intersection between the line and the parabola, we set the equations equal to each other:
\[
4x - 8 = (x - 2)^2 + 3.
\]
Expanding the quadratic:
\[
4x - 8 = x^2 - 4x + 4 + 3.
\]
Rearranging gives:
\[
0 = x^2 - 8x + 15.
\]
Factoring the quadratic:
\[
0 = (x - 3)(x - 5).
\]
This gives us:
\[
x = 3 \quad \text{and} \quad x = 5.
\]
### Finding Corresponding y-values:
1. For \( x = 3 \):
\[
y = 4(3) - 8 = 12 - 8 = 4 \quad \Rightarrow \quad (3, 4).
\]
2. For \( x = 5 \):
\[
y = 4(5) - 8 = 20 - 8 = 12 \quad \Rightarrow \quad (5, 12).
\]
### Conclusion:
The solutions to the system are the points of intersection, which are:
- \( (3, 4) \)
- \( (5, 12) \)
Thus, the correct response for the solutions to the system is:
**(3, 4) and (5, 12)**.