What is the graph of the equation?
y = –x2 + 2x + 3
(1 point)
Responses
graphThe function has a maximum at left-parenthesis 1 comma 4 right-parenthesis and passes through the points left-parenthesis 0 comma 3 right-parenthesis and left-parenthesis 2 comma 3 right-parenthesis.
Image with alt text: graph The function has a maximum at left-parenthesis 1 comma 4 right-parenthesis and passes through the points left-parenthesis 0 comma 3 right-parenthesis and left-parenthesis 2 comma 3 right-parenthesis.
graphThe function has a minimum at left-parenthesis 1 comma negative 4 right-parenthesis and passes through the points left-parenthesis 0 comma negative 3 right-parenthesis and left-parenthesis 2 comma negative 3 right-parenthesis.
Image with alt text: graph The function has a minimum at left-parenthesis 1 comma negative 4 right-parenthesis and passes through the points left-parenthesis 0 comma negative 3 right-parenthesis and left-parenthesis 2 comma negative 3 right-parenthesis.
graph
Image with alt text: graph
graphThe function has a minimum at left-parenthesis negative 1 comma negative 4 right-parenthesis and passes through the points left-parenthesis negative 2 comma negative 3 right-parenthesis and left-parenthesis 0 comma negative 3 right-parenthesis.
Image with alt text: graph The function has a minimum at left-parenthesis negative 1 comma negative 4 right-parenthesis and passes through the points left-parenthesis negative 2 comma negative 3 right-parenthesis and left-parenthesis 0 comma negative 3 right-parenthesis.
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The correct response is:
graphThe function has a maximum at left-parenthesis 1 comma 4 right-parenthesis and passes through the points left-parenthesis 0 comma 3 right-parenthesis and left-parenthesis 2 comma 3 right-parenthesis.
Image with alt text: graph The function has a maximum at left-parenthesis 1 comma 4 right-parenthesis and passes through the points left-parenthesis 0 comma 3 right-parenthesis and left-parenthesis 2 comma 3 right-parenthesis.
The graph of the equation is:
Image with alt text: graph The function has a maximum at (1, 4) and passes through the points (0, 3) and (2, 3).
To find the graph of the equation y = -x^2 + 2x + 3, you can follow these steps:
1. Determine the vertex of the parabola: The vertex of the parabola can be found using the formula x = -b / (2a), where a and b are the coefficients of the quadratic equation. In this case, a = -1 and b = 2. Plugging these values into the formula, we get x = -2 / (2 * -1) = 1.
2. Find the y-coordinate of the vertex: To find the y-coordinate, substitute the x-coordinate (1) into the equation. y = -(1)^2 + 2(1) + 3 = -1 + 2 + 3 = 4. So the vertex is (1, 4).
3. Determine the direction of the parabola: Since the coefficient of x^2 is negative (-1), the parabola opens downwards and has a maximum point.
4. Find the x-intercepts: To find the x-intercepts, set y = 0 and solve for x. 0 = -x^2 + 2x + 3. This equation can be solved using factoring, completing the square, or the quadratic formula. In this case, the equation cannot be factored easily, so the quadratic formula can be used: x = (-2 ± √(2^2 - 4(-1)(3))) / (2*-1) = (-2 ± √(4 + 12)) / -2 = (-2 ± √16) / -2 = (-2 ± 4) / -2. Solving this gives two x-intercepts, -2 and 3.
5. Determine if the parabola opens upwards or downwards: Since the coefficient of x^2 is negative (-1), the parabola opens downwards.
6. Plot the vertex, x-intercepts, and any additional points: Based on the above calculations, the graph of the equation y = -x^2 + 2x + 3 will have a maximum point at (1, 4), and it will pass through the x-intercepts (-2, 0) and (3, 0).
Therefore, the correct response is:
graph The function has a maximum at (1, 4) and passes through the points (0, 3) and (2, 3).