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Questions and Answers
Which of the following correctly defines the domain of a relation?
Which of the following correctly defines the domain of a relation?
What constitutes a function according to the definition provided?
What constitutes a function according to the definition provided?
When sketching the graph of a relation with inequalities, what is the first step?
When sketching the graph of a relation with inequalities, what is the first step?
In the provided relation A: R = {(x, y): y ≥ x and y ≥ -2x + 4}, what will the region look like?
In the provided relation A: R = {(x, y): y ≥ x and y ≥ -2x + 4}, what will the region look like?
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Which of the following is true about the range of a function?
Which of the following is true about the range of a function?
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If a relation has ordered pairs with the same first element but different second elements, what is it classified as?
If a relation has ordered pairs with the same first element but different second elements, what is it classified as?
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In the inequality relation B: R = {(x, y): x - 2y ≤ 0 and |x| ≥ 1}, which part is true?
In the inequality relation B: R = {(x, y): x - 2y ≤ 0 and |x| ≥ 1}, which part is true?
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What condition must be satisfied for a relation to be considered a function?
What condition must be satisfied for a relation to be considered a function?
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What is the y-intercept of the function defined by the equation f(x) = 3x^2 + 2?
What is the y-intercept of the function defined by the equation f(x) = 3x^2 + 2?
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In the quadratic function f(x) = ax^2 + bx + c, what can be stated about the coefficient 'a'?
In the quadratic function f(x) = ax^2 + bx + c, what can be stated about the coefficient 'a'?
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When is the x-intercept of a function f(x) found?
When is the x-intercept of a function f(x) found?
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Which of the following forms represents a quadratic function?
Which of the following forms represents a quadratic function?
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From the function f(x) = (x - 3)(x + 3), how would this be simplified in standard form?
From the function f(x) = (x - 3)(x + 3), how would this be simplified in standard form?
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What is the range of values for x if you are evaluating the function in the interval [-2, 2]?
What is the range of values for x if you are evaluating the function in the interval [-2, 2]?
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What shape does the graph of the quadratic function f(x) = -2x^2 take?
What shape does the graph of the quadratic function f(x) = -2x^2 take?
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Which of the following points represents the x-intercept of the function graphed by f(x) = 2x^2 + 4x + 2?
Which of the following points represents the x-intercept of the function graphed by f(x) = 2x^2 + 4x + 2?
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Which of the following pairs correctly represents a function?
Which of the following pairs correctly represents a function?
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What is the domain of the function R1 if given as R1 = {(1, a), (3, b), (5, a)}?
What is the domain of the function R1 if given as R1 = {(1, a), (3, b), (5, a)}?
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Which of the following best describes the range of the function R3 = {(1, a), (3, b), (5, c)}?
Which of the following best describes the range of the function R3 = {(1, a), (3, b), (5, c)}?
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What does f(x) represent in function notation?
What does f(x) represent in function notation?
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If a relation R is defined as R = {(x, y): y is the mother of x}, what is true about this relation?
If a relation R is defined as R = {(x, y): y is the mother of x}, what is true about this relation?
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Which statement is true about the function notation f(x) = 2x + 1?
Which statement is true about the function notation f(x) = 2x + 1?
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If a relation pairs 7 with both 8 and 6, what can be concluded about it?
If a relation pairs 7 with both 8 and 6, what can be concluded about it?
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In the function f(x) = x - 3, what is the functional value at x = 5?
In the function f(x) = x - 3, what is the functional value at x = 5?
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What are the values of the function when x is -1 for the function f(x) = 2x^2 + 3?
What are the values of the function when x is -1 for the function f(x) = 2x^2 + 3?
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What is the range of the function f(x) = 2x^2 + 3?
What is the range of the function f(x) = 2x^2 + 3?
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What is the vertex of the function f(x) = -2x^2 + 3?
What is the vertex of the function f(x) = -2x^2 + 3?
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Which statement is true regarding the graph of the function when a > 0?
Which statement is true regarding the graph of the function when a > 0?
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What happens to the axis of symmetry for the function f(x) = 2x^2 + c as c increases?
What happens to the axis of symmetry for the function f(x) = 2x^2 + c as c increases?
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For the function f(x) = 2x^2 - 3, what is the value of f(2)?
For the function f(x) = 2x^2 - 3, what is the value of f(2)?
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Which graph characteristic is true for the function f(x) = -2x^2 - 3?
Which graph characteristic is true for the function f(x) = -2x^2 - 3?
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What is the effect of changing the coefficient a from positive to negative in the function f(x) = ax^2 + c?
What is the effect of changing the coefficient a from positive to negative in the function f(x) = ax^2 + c?
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What is the highest or lowest point of a parabola called?
What is the highest or lowest point of a parabola called?
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What does the axis of symmetry of a parabola represent?
What does the axis of symmetry of a parabola represent?
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If a quadratic function is defined by the equation $f(x) = a(x - h)^2 + k$ and $a < 0$, what can be concluded about the graph?
If a quadratic function is defined by the equation $f(x) = a(x - h)^2 + k$ and $a < 0$, what can be concluded about the graph?
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What happens to the graph of a quadratic function when the absolute value of $a$ increases?
What happens to the graph of a quadratic function when the absolute value of $a$ increases?
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When the graph of a quadratic function opens upward, what type of value does the vertex represent?
When the graph of a quadratic function opens upward, what type of value does the vertex represent?
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In the equation $f(x) = a(x - h)^2 + k$, how do you determine the direction the parabola opens?
In the equation $f(x) = a(x - h)^2 + k$, how do you determine the direction the parabola opens?
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If a quadratic function is given by $f(x) = 2x^2 + 3$, what is the direction of its graph?
If a quadratic function is given by $f(x) = 2x^2 + 3$, what is the direction of its graph?
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Shifting the graph of $f(x) = ax^2$ horizontally to the right is achieved by which transformation?
Shifting the graph of $f(x) = ax^2$ horizontally to the right is achieved by which transformation?
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Study Notes
Relations
- A relation is a set of ordered pairs.
- The domain of a relation is the set of all first elements of the ordered pairs.
- The range of a relation is the set of all second elements of the ordered pairs.
- A relation is a function if and only if no two ordered pairs have the same first element.
Functions
- A function is a relation in which no two ordered pairs have the same first element.
- The domain of a function is the set of all possible input values.
- The range of a function is the set of all possible output values.
- The functional value of f at x is denoted by f(x) and is called the image of x under the function f.
Graphing Relations
- To sketch the graph of a relation with two or more inequalities, sketch the regions of each inequality on the same coordinate system.
- The intersection of the regions is the graph of the relation.
Graphs of Linear Functions
- The graph of a linear function is a straight line.
- The slope-intercept form of a linear function is y = mx + b, where m is the slope and b is the y-intercept.
- The y-intercept is the point where the graph crosses the y-axis.
- The x-intercept is the point where the graph crosses the x-axis.
Graphs of Quadratic Functions
- A quadratic function is a function defined by f(x) = ax^2 + bx + c, where a, b, and c are real numbers and a ≠ 0.
- The graph of a quadratic function is a parabola.
- The vertex is the lowest or highest point of the parabola.
- The axis of symmetry is the vertical line that passes through the vertex.
- If a > 0, the parabola opens upward.
- If a < 0, the parabola opens downward.
- The graph of f(x) = a(x - h)^2 + k can be obtained by shifting the graph of f(x) = ax^2, h units to the right if h > 0, and h units to the left if h < 0, and k units up if k > 0, and k units down if k < 0.
Minimum and Maximum Values of Quadratic Functions
- If the graph of a quadratic function opens upward, the function has a minimum value.
- If the graph of a quadratic function opens downward, the function has a maximum value.
- The minimum or maximum value of a quadratic function is obtained at the vertex of its graph.
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Description
Test your understanding of relations and functions, including their definitions, domains, ranges, and the characteristics that distinguish a function from a general relation. Additionally, explore how to graph these relations and linear functions effectively.
