Question
What is the graph of the equation?
y = x2 – 4x + 3
y = x2 – 4x + 3
Answers
GPT 3.5
The graph of the equation y = x^2 - 4x + 3 is a parabola that opens upward. To determine the shape and position of the parabola, we can examine the coefficients of the terms:
- The coefficient of the x^2 term is positive (1), indicating that the parabola opens upward.
- The constant term is positive (3), which means the parabola intersects the y-axis at y = 3.
Next, we can find the vertex of the parabola using the formula x = -b/2a. In this equation, a = 1 and b = -4:
x = -(-4)/(2*1) = 4/2 = 2
So, the x-coordinate of the vertex is 2. To find the y-coordinate, we substitute this x-value back into the equation:
y = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1
Therefore, the vertex of the parabola is (2, -1). As the coefficient of the x^2 term is positive, the parabola opens upward from this vertex.
To plot additional points, we can use the symmetry of the parabola. Since the x-coordinate of the vertex is 2, the same distance in either direction from the vertex gives us points on the graph.
For example, if we move one unit to the right from the vertex (2, -1), we get the point (3, 2). If we move one unit to the left from the vertex, we get the point (1, 2). We can plot these points and the y-intercept (0, 3) to get a better sense of the shape of the parabola.
Here is a rough sketch of the graph of the equation y = x^2 - 4x + 3:
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-1 0 1 2 3 4 5 6 7 8 9 x
- The coefficient of the x^2 term is positive (1), indicating that the parabola opens upward.
- The constant term is positive (3), which means the parabola intersects the y-axis at y = 3.
Next, we can find the vertex of the parabola using the formula x = -b/2a. In this equation, a = 1 and b = -4:
x = -(-4)/(2*1) = 4/2 = 2
So, the x-coordinate of the vertex is 2. To find the y-coordinate, we substitute this x-value back into the equation:
y = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1
Therefore, the vertex of the parabola is (2, -1). As the coefficient of the x^2 term is positive, the parabola opens upward from this vertex.
To plot additional points, we can use the symmetry of the parabola. Since the x-coordinate of the vertex is 2, the same distance in either direction from the vertex gives us points on the graph.
For example, if we move one unit to the right from the vertex (2, -1), we get the point (3, 2). If we move one unit to the left from the vertex, we get the point (1, 2). We can plot these points and the y-intercept (0, 3) to get a better sense of the shape of the parabola.
Here is a rough sketch of the graph of the equation y = x^2 - 4x + 3:
5 |
| *
|
|
|
| *
| *
|
| *
|
0 |__*___*___*___*___*___*___*___*___*___*_
-1 0 1 2 3 4 5 6 7 8 9 x