Q. According to the passage, what is the significance of the vertex in a quadratic function?
A.
It represents the function's maximum or minimum value.
B.
It is the point where the function crosses the y-axis.
C.
It indicates the function's slope.
D.
It is the point of discontinuity.
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Solution
The vertex of a quadratic function is crucial as it represents the maximum or minimum point of the graph.
Correct Answer:
A
— It represents the function's maximum or minimum value.
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Q. Based on the passage, which of the following statements about the graph of a quadratic function is true?
A.
It can have at most one x-intercept.
B.
It is always increasing.
C.
It is a parabola that opens upwards or downwards.
D.
It has no maximum or minimum points.
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Solution
A quadratic function is represented by a parabola, which can open upwards or downwards depending on the coefficient of the squared term.
Correct Answer:
C
— It is a parabola that opens upwards or downwards.
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Q. If a function f is defined as f(x) = 3x + 2, what is the value of f(4)?
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Solution
To find f(4), substitute x with 4: f(4) = 3(4) + 2 = 12 + 2 = 14.
Correct Answer:
A
— 14
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Q. If a function f(x) is defined as f(x) = 2x + 3, what is the slope of its graph?
A.
0
B.
2
C.
3
D.
Undefined
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Solution
In the linear function f(x) = mx + b, 'm' represents the slope, which is 2 in this case.
Correct Answer:
B
— 2
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Q. If a function f(x) is defined as f(x) = 2x + 3, what is the slope of the graph of this function?
A.
0
B.
2
C.
3
D.
Undefined
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Solution
In the linear function f(x) = 2x + 3, the coefficient of x (which is 2) represents the slope of the graph.
Correct Answer:
B
— 2
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Q. If a function f(x) is defined as f(x) = 2x + 3, what is the slope of the graph?
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Solution
In the linear function f(x) = mx + b, 'm' represents the slope. Here, the slope is 2.
Correct Answer:
C
— 2
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Q. If a function f(x) is defined as f(x) = 2x + 3, what is the value of f(0)?
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Solution
Substituting x = 0 into the function gives f(0) = 2(0) + 3 = 3.
Correct Answer:
C
— 3
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Q. If a function f(x) is defined as f(x) = 2x + 5, what is the slope of the graph?
A.
0
B.
2
C.
5
D.
Undefined
Show solution
Solution
In the linear function f(x) = mx + b, 'm' represents the slope. Here, m = 2.
Correct Answer:
B
— 2
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Q. If a function f(x) is defined as f(x) = 3x + 2, what is the value of f(4)?
Show solution
Solution
To find f(4), substitute x with 4: f(4) = 3(4) + 2 = 12 + 2 = 14.
Correct Answer:
A
— 14
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Q. If a function f(x) is defined as f(x) = 3x - 5, what is the slope of the graph of this function?
A.
3
B.
-5
C.
0
D.
Undefined
Show solution
Solution
In the linear function f(x) = mx + b, m represents the slope. Here, m = 3, so the slope is 3.
Correct Answer:
A
— 3
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Q. If a function f(x) is defined as f(x) = 3x - 5, what is the slope of the graph?
A.
3
B.
-5
C.
0
D.
Undefined
Show solution
Solution
In the linear function f(x) = mx + b, 'm' represents the slope, which is 3 in this case.
Correct Answer:
A
— 3
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Q. If a function f(x) is defined as f(x) = x^3 - 3x + 2, what can be inferred about its behavior at critical points?
A.
It has no critical points.
B.
It has one local maximum and one local minimum.
C.
It is always increasing.
D.
It is always decreasing.
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Solution
To find critical points, we take the derivative and set it to zero. The function has one local maximum and one local minimum based on the nature of cubic functions.
Correct Answer:
B
— It has one local maximum and one local minimum.
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Q. If a function f(x) is defined as f(x) = x^3 - 3x, what is the nature of its critical points?
A.
They can be local maxima, local minima, or points of inflection.
B.
They are always local maxima.
C.
They are always local minima.
D.
They do not exist.
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Solution
Critical points of a function can be local maxima, local minima, or points of inflection, depending on the behavior of the function around those points.
Correct Answer:
A
— They can be local maxima, local minima, or points of inflection.
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Q. If a function is defined as f(x) = 3x + 2, what is the slope of the line represented by this function?
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Solution
In the linear function f(x) = mx + b, 'm' represents the slope. Here, the slope is 3.
Correct Answer:
A
— 3
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Q. If the derivative of a function f(x) is positive for all x in its domain, what can be inferred about the function?
A.
The function is decreasing.
B.
The function is constant.
C.
The function is increasing.
D.
The function has a maximum point.
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Solution
If the derivative of a function is positive for all x, it indicates that the function is increasing throughout its domain.
Correct Answer:
C
— The function is increasing.
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Q. If the function f(x) is defined as f(x) = 2x + 1, what is the value of f(3)?
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Solution
Substituting x = 3 into the function gives f(3) = 2(3) + 1 = 6 + 1 = 7.
Correct Answer:
C
— 7
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Q. If the function g(x) = 2x + 3 is transformed to g(x) = 2(x - 1) + 3, what type of transformation has occurred?
A.
Vertical shift up.
B.
Vertical shift down.
C.
Horizontal shift left.
D.
Horizontal shift right.
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Solution
The transformation g(x) = 2(x - 1) + 3 indicates a horizontal shift to the right by 1 unit.
Correct Answer:
D
— Horizontal shift right.
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Q. If the graph of a function f(x) intersects the x-axis at x = 1 and x = 3, which of the following can be inferred?
A.
f(1) = 0 and f(3) = 0.
B.
The function is linear.
C.
The function has no real roots.
D.
The function is increasing.
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Solution
If the graph intersects the x-axis at x = 1 and x = 3, it means that f(1) = 0 and f(3) = 0, indicating the roots of the function.
Correct Answer:
A
— f(1) = 0 and f(3) = 0.
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Q. If the graph of a function f(x) intersects the x-axis at x = 3, what can be concluded?
A.
f(3) = 0.
B.
f(3) > 0.
C.
f(3) < 0.
D.
f(3) is undefined.
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Solution
The intersection of the graph with the x-axis indicates that the function value at that point is zero.
Correct Answer:
A
— f(3) = 0.
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Q. If the graph of a function f(x) is symmetric about the y-axis, which of the following must be true?
A.
f(x) = f(-x) for all x.
B.
f(x) = -f(-x) for all x.
C.
f(x) is always positive.
D.
f(x) has a maximum value.
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Solution
A function that is symmetric about the y-axis satisfies the property f(x) = f(-x) for all x.
Correct Answer:
A
— f(x) = f(-x) for all x.
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Q. If the graph of a function is a parabola opening upwards, which of the following can be inferred about the function?
A.
The function has a maximum value.
B.
The function has a minimum value.
C.
The function is linear.
D.
The function is constant.
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Solution
A parabola that opens upwards indicates that the function has a minimum value at its vertex.
Correct Answer:
B
— The function has a minimum value.
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Q. If the graph of a function is symmetric about the y-axis, which of the following types of functions could it represent?
A.
Linear function
B.
Odd function
C.
Even function
D.
Exponential function
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Solution
A function is symmetric about the y-axis if it is an even function, which satisfies the condition f(x) = f(-x).
Correct Answer:
C
— Even function
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Q. If the graph of a function is symmetric about the y-axis, which of the following types of functions could it be?
A.
Linear function
B.
Odd function
C.
Even function
D.
Exponential function
Show solution
Solution
A function is symmetric about the y-axis if it is an even function, which satisfies the condition f(x) = f(-x).
Correct Answer:
C
— Even function
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Q. If the graph of a function is symmetric about the y-axis, which of the following must be true?
A.
The function is linear.
B.
The function is even.
C.
The function is odd.
D.
The function has no intercepts.
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Solution
A function is even if it is symmetric about the y-axis, meaning f(x) = f(-x) for all x.
Correct Answer:
B
— The function is even.
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Q. In a function f(x) = ax^2 + bx + c, if a > 0, what can be inferred about the direction of the graph?
A.
The graph opens upwards.
B.
The graph opens downwards.
C.
The graph is a straight line.
D.
The graph is a constant function.
Show solution
Solution
When a > 0 in a quadratic function, the graph opens upwards, indicating that the vertex is the minimum point.
Correct Answer:
A
— The graph opens upwards.
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Q. In a function f(x) = ax^2 + bx + c, if a > 0, what can be said about the graph of the function?
A.
It opens upwards.
B.
It opens downwards.
C.
It has a maximum point.
D.
It is a straight line.
Show solution
Solution
If a > 0 in a quadratic function, the graph opens upwards, indicating that it has a minimum point.
Correct Answer:
A
— It opens upwards.
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Q. In a function f(x) = ax^2 + bx + c, what does the coefficient 'a' determine about the graph?
A.
The y-intercept of the graph.
B.
The direction of the parabola's opening.
C.
The x-intercepts of the graph.
D.
The slope of the graph.
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Solution
'a' determines the direction of the parabola's opening; if 'a' is positive, it opens upwards, and if negative, it opens downwards.
Correct Answer:
B
— The direction of the parabola's opening.
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Q. In a function f(x) = ax^2 + bx + c, what does the coefficient 'a' determine?
A.
The direction of the parabola's opening.
B.
The y-intercept of the graph.
C.
The slope of the graph.
D.
The x-intercepts of the graph.
Show solution
Solution
The coefficient 'a' in a quadratic function determines whether the parabola opens upwards (a > 0) or downwards (a < 0).
Correct Answer:
A
— The direction of the parabola's opening.
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Q. In a function f(x) = ax^2 + bx + c, what does the value of 'a' determine about the graph?
A.
The y-intercept of the graph.
B.
The direction of the parabola.
C.
The x-intercepts of the graph.
D.
The maximum value of the function.
Show solution
Solution
'a' determines the direction of the parabola; if 'a' is positive, it opens upwards, and if negative, it opens downwards.
Correct Answer:
B
— The direction of the parabola.
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Q. In a function f(x) = ax^2 + bx + c, what does the value of 'a' determine?
A.
The direction in which the parabola opens.
B.
The x-intercepts of the graph.
C.
The y-intercept of the graph.
D.
The maximum value of the function.
Show solution
Solution
'a' determines the direction of the parabola; if 'a' is positive, it opens upwards, and if negative, it opens downwards.
Correct Answer:
A
— The direction in which the parabola opens.
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