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The function f is defined by the equation f (x) = x - x2. Which of the following represents a quadratic with no real zeros?
- a)
- b)
- c)
- d)
Correct answer is option 'B'. Can you explain this answer?
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The function f is defined by the equation f (x) = x - x2. Which of the...
Perhaps the simplest way to begin this problem is to draw a quick sketch of the function in the xy-plane, and then compare this graph to the transformations of the original function given in the choices. Notice that the original function f(x) = x - x2 is easily factored as f(x) = x (1 - x). The Zero Product Property tells us that this function must have zeros at x = 0 and x = 1. Notice, also, that since the coefficient of the x2 term in the original function is negative (-1), the graph of this quadratic is an “open-down” parabola. Also, the axis of symmetry is halfway between the zeros, at x = 1/2. Plugging x = 1/2 back into the function gives us 
and therefore, the vertex of the parabola is 
The question asks us to find the function that has no real zeros. This means that the graph of this function must not intersect the x-axis at all. Each answer choice indicates a different transformation of the function f, that choice (A)f(x) + 1/2 is the graph off shifted up 1/2 unit, choice (B) f(x) - 1/2 is the graph of f shifted down 1/2 unit, choice (C) f(x/2) is the graph off stretched by a factor of 2 in the horizontal direction, and choice (D) f(x - 1/2) is the graph off shifted right 1/2 unit.
As the sketch above shows, only (B) yields a graph that does not intersect the x-axis.
and therefore, the vertex of the parabola is The question asks us to find the function that has no real zeros. This means that the graph of this function must not intersect the x-axis at all. Each answer choice indicates a different transformation of the function f, that choice (A)f(x) + 1/2 is the graph off shifted up 1/2 unit, choice (B) f(x) - 1/2 is the graph of f shifted down 1/2 unit, choice (C) f(x/2) is the graph off stretched by a factor of 2 in the horizontal direction, and choice (D) f(x - 1/2) is the graph off shifted right 1/2 unit.
As the sketch above shows, only (B) yields a graph that does not intersect the x-axis.
Most Upvoted Answer
The function f is defined by the equation f (x) = x - x2. Which of the...
Perhaps the simplest way to begin this problem is to draw a quick sketch of the function in the xy-plane, and then compare this graph to the transformations of the original function given in the choices. Notice that the original function f(x) = x - x2 is easily factored as f(x) = x (1 - x). The Zero Product Property tells us that this function must have zeros at x = 0 and x = 1. Notice, also, that since the coefficient of the x2 term in the original function is negative (-1), the graph of this quadratic is an “open-down” parabola. Also, the axis of symmetry is halfway between the zeros, at x = 1/2. Plugging x = 1/2 back into the function gives us and therefore, the vertex of the parabola is
The question asks us to find the function that has no real zeros. This means that the graph of this function must not intersect the x-axis at all. Each answer choice indicates a different transformation of the function f, that choice (A)f(x) + 1/2 is the graph off shifted up 1/2 unit, choice (B) f(x) - 1/2 is the graph of f shifted down 1/2 unit, choice (C) f(x/2) is the graph off stretched by a factor of 2 in the horizontal direction, and choice (D) f(x - 1/2) is the graph off shifted right 1/2 unit.
As the sketch above shows, only (B) yields a graph that does not intersect the x-axis.
and therefore, the vertex of the parabola is The question asks us to find the function that has no real zeros. This means that the graph of this function must not intersect the x-axis at all. Each answer choice indicates a different transformation of the function f, that choice (A)f(x) + 1/2 is the graph off shifted up 1/2 unit, choice (B) f(x) - 1/2 is the graph of f shifted down 1/2 unit, choice (C) f(x/2) is the graph off stretched by a factor of 2 in the horizontal direction, and choice (D) f(x - 1/2) is the graph off shifted right 1/2 unit.
As the sketch above shows, only (B) yields a graph that does not intersect the x-axis.
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The function f is defined by the equation f (x) = x - x2. Which of the following represents a quadratic with no real zeros?a)b)c)d)Correct answer is option 'B'. Can you explain this answer?
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