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The graph of the equation y = x2 + k in the xy-plane is a parabola with a vertex that is below the x-axis. Which of the following is true of the parabola represented by the equation y = k(x - b)2 - c?
- a)The vertex is (b, -c), and the parabola opens downward.
- b)The vertex is (b, -c), and the parabola opens upward.
- c)The vertex is (-b, c), and the parabola opens downward.
- d)The vertex is (-b, c), and the parabola opens upward.
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
The graph of the equation y = x2 + k in the xy-plane is a parabola wit...
An equation in the form y = a(x - h)2 + k represents a parabola in the xy-plane, with vertex (h, k). If a is positive, the parabola opens upward, and if a is negative it opens downward. Therefore the parabola represented by y = x2 + k has vertex (0, k) and is open up. If this vertex is below the x-axis, then k must be negative, and therefore y = k(x - b)2 - c represents a parabola with vertex (b, -c) that is open downward.
Most Upvoted Answer
The graph of the equation y = x2 + k in the xy-plane is a parabola wit...
Explanation:
To determine the properties of the parabola represented by the equation y = k(x - b)² - c, we can compare it to the standard form of a parabola equation, y = a(x - h)² + k, where (h, k) represents the vertex of the parabola.
1. Vertex:
The vertex of the parabola represented by the given equation is (b, -c). This can be determined by comparing the equation to the standard form. In the given equation, the term (x - b)² indicates a horizontal shift of b units to the right, and the term -c indicates a vertical shift of c units downward. Therefore, the vertex of the parabola is at the point (b, -c).
2. Orientation:
The coefficient of the quadratic term, k, determines the orientation of the parabola.
- If k > 0, the parabola opens upward.
- If k < 0,="" the="" parabola="" opens="" />
In the given equation, the coefficient k is multiplied by (x - b)². Since the given equation is y = k(x - b)² - c, we can see that k is being multiplied by a positive value of (x - b)². Therefore, the sign of k determines the orientation of the parabola.
Combining the information:
- The vertex of the parabola represented by the given equation is (b, -c).
- The coefficient k is positive, indicating that the parabola opens upward.
Conclusion:
Therefore, the correct statement about the parabola represented by the equation y = k(x - b)² - c is:
a) The vertex is (b, -c), and the parabola opens downward.
To determine the properties of the parabola represented by the equation y = k(x - b)² - c, we can compare it to the standard form of a parabola equation, y = a(x - h)² + k, where (h, k) represents the vertex of the parabola.
1. Vertex:
The vertex of the parabola represented by the given equation is (b, -c). This can be determined by comparing the equation to the standard form. In the given equation, the term (x - b)² indicates a horizontal shift of b units to the right, and the term -c indicates a vertical shift of c units downward. Therefore, the vertex of the parabola is at the point (b, -c).
2. Orientation:
The coefficient of the quadratic term, k, determines the orientation of the parabola.
- If k > 0, the parabola opens upward.
- If k < 0,="" the="" parabola="" opens="" />
In the given equation, the coefficient k is multiplied by (x - b)². Since the given equation is y = k(x - b)² - c, we can see that k is being multiplied by a positive value of (x - b)². Therefore, the sign of k determines the orientation of the parabola.
Combining the information:
- The vertex of the parabola represented by the given equation is (b, -c).
- The coefficient k is positive, indicating that the parabola opens upward.
Conclusion:
Therefore, the correct statement about the parabola represented by the equation y = k(x - b)² - c is:
a) The vertex is (b, -c), and the parabola opens downward.
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The graph of the equation y = x2 + k in the xy-plane is a parabola with a vertex that is below the x-axis. Which of the following is true of the parabola represented by the equation y = k(x - b)2 - c?a)The vertex is (b, -c), and the parabola opens downward.b)The vertex is (b, -c), and the parabola opens upward.c)The vertex is (-b, c), and the parabola opens downward.d)The vertex is (-b, c), and the parabola opens upward.Correct answer is option 'A'. Can you explain this answer?
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The graph of the equation y = x2 + k in the xy-plane is a parabola with a vertex that is below the x-axis. Which of the following is true of the parabola represented by the equation y = k(x - b)2 - c?a)The vertex is (b, -c), and the parabola opens downward.b)The vertex is (b, -c), and the parabola opens upward.c)The vertex is (-b, c), and the parabola opens downward.d)The vertex is (-b, c), and the parabola opens upward.Correct answer is option 'A'. Can you explain this answer? for SAT 2025 is part of SAT preparation. The Question and answers have been prepared according to the SAT exam syllabus. Information about The graph of the equation y = x2 + k in the xy-plane is a parabola with a vertex that is below the x-axis. Which of the following is true of the parabola represented by the equation y = k(x - b)2 - c?a)The vertex is (b, -c), and the parabola opens downward.b)The vertex is (b, -c), and the parabola opens upward.c)The vertex is (-b, c), and the parabola opens downward.d)The vertex is (-b, c), and the parabola opens upward.Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for SAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The graph of the equation y = x2 + k in the xy-plane is a parabola with a vertex that is below the x-axis. Which of the following is true of the parabola represented by the equation y = k(x - b)2 - c?a)The vertex is (b, -c), and the parabola opens downward.b)The vertex is (b, -c), and the parabola opens upward.c)The vertex is (-b, c), and the parabola opens downward.d)The vertex is (-b, c), and the parabola opens upward.Correct answer is option 'A'. Can you explain this answer?.
The graph of the equation y = x2 + k in the xy-plane is a parabola with a vertex that is below the x-axis. Which of the following is true of the parabola represented by the equation y = k(x - b)2 - c?a)The vertex is (b, -c), and the parabola opens downward.b)The vertex is (b, -c), and the parabola opens upward.c)The vertex is (-b, c), and the parabola opens downward.d)The vertex is (-b, c), and the parabola opens upward.Correct answer is option 'A'. Can you explain this answer? for SAT 2025 is part of SAT preparation. The Question and answers have been prepared according to the SAT exam syllabus. Information about The graph of the equation y = x2 + k in the xy-plane is a parabola with a vertex that is below the x-axis. Which of the following is true of the parabola represented by the equation y = k(x - b)2 - c?a)The vertex is (b, -c), and the parabola opens downward.b)The vertex is (b, -c), and the parabola opens upward.c)The vertex is (-b, c), and the parabola opens downward.d)The vertex is (-b, c), and the parabola opens upward.Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for SAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The graph of the equation y = x2 + k in the xy-plane is a parabola with a vertex that is below the x-axis. Which of the following is true of the parabola represented by the equation y = k(x - b)2 - c?a)The vertex is (b, -c), and the parabola opens downward.b)The vertex is (b, -c), and the parabola opens upward.c)The vertex is (-b, c), and the parabola opens downward.d)The vertex is (-b, c), and the parabola opens upward.Correct answer is option 'A'. Can you explain this answer?.
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