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Questions and Answers
What are the coordinates of the origin in a cartesian coordinate system?
What are the coordinates of the origin in a cartesian coordinate system?
Which axis is typically oriented vertically in the Cartesian coordinate system?
Which axis is typically oriented vertically in the Cartesian coordinate system?
In terms of quadrant designation, which quadrant contains points where both coordinates are positive?
In terms of quadrant designation, which quadrant contains points where both coordinates are positive?
What term describes the horizontal line in the Cartesian coordinate system?
What term describes the horizontal line in the Cartesian coordinate system?
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The points (1, 0) and (0, 1) are located on which part of the Cartesian coordinate system?
The points (1, 0) and (0, 1) are located on which part of the Cartesian coordinate system?
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Which of the following best describes the coordinate plane?
Which of the following best describes the coordinate plane?
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Which point represents a location on the negative X-axis?
Which point represents a location on the negative X-axis?
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What is the primary purpose of the rectangular coordinate system?
What is the primary purpose of the rectangular coordinate system?
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What does a positive slope indicate about the behavior of a linear function's graph?
What does a positive slope indicate about the behavior of a linear function's graph?
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What are the possible values for the domain and range of any linear function f(x) = mx + b, where m ≠ 0?
What are the possible values for the domain and range of any linear function f(x) = mx + b, where m ≠ 0?
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Which statement correctly describes the x-intercept of a linear function?
Which statement correctly describes the x-intercept of a linear function?
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What effect does a larger absolute value of the slope (|m|) have on the graph of a linear function?
What effect does a larger absolute value of the slope (|m|) have on the graph of a linear function?
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What is true about the graph of a linear function regarding its continuity?
What is true about the graph of a linear function regarding its continuity?
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As x approaches positive infinity, what happens to f(x) for a linear function with a negative slope?
As x approaches positive infinity, what happens to f(x) for a linear function with a negative slope?
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What does the y-intercept of a linear function represent?
What does the y-intercept of a linear function represent?
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If a linear function has a slope of zero, what is true about its graph?
If a linear function has a slope of zero, what is true about its graph?
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Which statement accurately describes when a function is considered a one-to-one correspondence?
Which statement accurately describes when a function is considered a one-to-one correspondence?
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In the context of function graphing, what does the x-intercept signify?
In the context of function graphing, what does the x-intercept signify?
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Which of the following functions is not onto when mapped from ℝ to ℝ?
Which of the following functions is not onto when mapped from ℝ to ℝ?
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Given the function defined as f(x) = 2x + 3, which of the following statements is true?
Given the function defined as f(x) = 2x + 3, which of the following statements is true?
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What characterizes the behavior of a linear function graph?
What characterizes the behavior of a linear function graph?
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Which of the following statements is false regarding the range of a function?
Which of the following statements is false regarding the range of a function?
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Which of the following represents a constant function?
Which of the following represents a constant function?
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When checking if the function f(x) = 2x is one-to-one, what conclusion can be drawn?
When checking if the function f(x) = 2x is one-to-one, what conclusion can be drawn?
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What occurs at the vertex of a quadratic function?
What occurs at the vertex of a quadratic function?
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What is the characteristic of the quadratic function $f(x) = -2x^2 + 3x - 1$?
What is the characteristic of the quadratic function $f(x) = -2x^2 + 3x - 1$?
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How can the extreme values of a quadratic function be found?
How can the extreme values of a quadratic function be found?
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In the standard form of a quadratic function $f(x) = ax^2 + bx + c$, what indicates that the function has a minimum point?
In the standard form of a quadratic function $f(x) = ax^2 + bx + c$, what indicates that the function has a minimum point?
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What is the result if a quadratic function has a positive leading coefficient?
What is the result if a quadratic function has a positive leading coefficient?
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What information can be derived directly from the vertex of a parabola defined by the function $y = ax^2 + bx + c$?
What information can be derived directly from the vertex of a parabola defined by the function $y = ax^2 + bx + c$?
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To find the optimal value of a quadratic function in standard form, what is the first step?
To find the optimal value of a quadratic function in standard form, what is the first step?
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What is the domain of the quadratic function $y = -x^2 + 6x - 8$?
What is the domain of the quadratic function $y = -x^2 + 6x - 8$?
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What does the vertex of a quadratic equation represent?
What does the vertex of a quadratic equation represent?
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If a quadratic equation has a graph that touches the x-axis but does not cross it, how many real solutions does it have?
If a quadratic equation has a graph that touches the x-axis but does not cross it, how many real solutions does it have?
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What is the first step to solve a quadratic equation graphically?
What is the first step to solve a quadratic equation graphically?
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When solving the inequality x² - 6x + 8 < 0 graphically, what interval represents the solution?
When solving the inequality x² - 6x + 8 < 0 graphically, what interval represents the solution?
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What can be inferred if the graph of a quadratic function does not touch or cross the x-axis?
What can be inferred if the graph of a quadratic function does not touch or cross the x-axis?
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For the quadratic function f(x) = x² + 4x + 8, what would the concavity of the graph indicate?
For the quadratic function f(x) = x² + 4x + 8, what would the concavity of the graph indicate?
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Which of the following describes the x-intercepts of a quadratic function?
Which of the following describes the x-intercepts of a quadratic function?
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In the quadratic function g(x) = -x² + 6x - 9, what characteristic does the negative leading coefficient indicate?
In the quadratic function g(x) = -x² + 6x - 9, what characteristic does the negative leading coefficient indicate?
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What condition must be met for a function to be classified as one-to-one?
What condition must be met for a function to be classified as one-to-one?
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Which method can be used to visually determine if a function is one-to-one?
Which method can be used to visually determine if a function is one-to-one?
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What defines an onto function?
What defines an onto function?
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Which of the following statements about the function $f(x) = x^2$ is true?
Which of the following statements about the function $f(x) = x^2$ is true?
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How can you prove that a function is not onto?
How can you prove that a function is not onto?
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Which function is an example of a one-to-one function?
Which function is an example of a one-to-one function?
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What conclusion can be drawn if a function graph intersects a horizontal line more than once?
What conclusion can be drawn if a function graph intersects a horizontal line more than once?
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For the function $f(x) = -2$, what can be said about its one-to-oneness and onto-ness?
For the function $f(x) = -2$, what can be said about its one-to-oneness and onto-ness?
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Study Notes
Grade 10 Mathematics Lecture Notes
- Course: Grade 10 Mathematics
- Instructor: Lemessa Olikà Kitìl (MA)
- Academic Year: 2017/2024
Unit 1: Relations and Functions
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1.1 Relations:
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1.1.1 Revision of Patterns:
- Patterns are identified through repeated arrangements of numbers, shapes, or colors.
- A number pattern is a sequence of numbers that follows a particular rule (arithmetic, geometric, or algebraic).
- Arithmetic patterns involve adding or subtracting a constant value to find the next term, while geometric patterns involve multiplying or dividing.
- Algebraic patterns involve finding a rule that defines the sequence.
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Activity 1.1:
- What is a pattern? A pattern is a recurring feature or design.
- Write the numbers that come next in 1, 3, 5, 7, 9 (11, 13).
- Identify the correct arrow from Figure 1.1 to fill in missing spaces.
- Construct a table showing values for the algebraic expression 2n + 3 (table 1.1).
- Determine the formula for Table 1.2 (Output = 3n + 2).
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1.1.1 Revision of Patterns:
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1.1.2 Cartesian Coordinate System in two Dimensions: - Also known as a rectangular coordinate system. - Defined by an ordered pair of perpendicular lines (axes). - Uses a single unit of length for both axes & has an orientation/location. - The axes meet at the origin (turning point). - A line drawn through a point perpendicular to each axis determines the coordinate. - The first coordinate is abscissa, and the second is ordinate—written as (a, b). - Origin has coordinates (0,0), while points equidistant from origin are (1, 0) and (0, 1). - The coordinate system divides the plane into quadrants (I, II, III, IV) as shown in Figure 1.3. - The signs of coordinates in each quadrant are shown (Table 1.1). - Cartesian coordinate system is used to locate points uniquely using x and y coordinates of points. - Two perpendicular directed lines (x and y axis) are specified, along with a unit length, to define the coordinates.
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1.1.3 Basic Concepts of Relation:
- Definition: A relation is a set of ordered pairs.
- Ordered pairs in a relation are not always numerical.
- Example exercises:
- Identify ordered pairs that meet a given relation definition.
- Given a set of ordered pairs, find the relation.
- Find a missing value in an ordered pair.
- Identify ordered pairs that meet a given relation definition.
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1.1.4 Graphs of Relations:
- Relations involving inequalities are graphed on a two-dimensional plane.
- The graph of the boundary line for inequalities should be a solid line for ≤ or ≥, and a broken/dashed line for < or >.
- Examples illustrate shading a graph reflecting a relation in the coordinate plane (Activity 1.4)
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1.2 Functions:
- Definition A function is a special type of relation.
- In a function, every input is paired with exactly one output.
- Determining whether relations represent functions or not.
- Using arrow diagrams in venn diagram to represent relations as functions.
- Example finding the domain and range
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Domain, Codomain and Range of a Function:
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Domain: All input values for which the function(fx) is defined.
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Codomain: Set of all possible output values.
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Range: Set of actual outputs which are pairs (x,y)
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1.2.2 Combinations of Functions: - Combining functions with +− operation, multiplication, division (exercise 1.9 & 1.10)
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1.2.3 Types of Functions:
- One-to-one (injective) functions have distinct images for different inputs.
- A horizontal line test can determine whether a graph represents a one-to-one function.
- Analyzing example functions to determine if they are one-to one (or not)
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1.3 Applications of Relations and Functions:
- Real-world applications of relations and functions are presented with examples
- Area of a square, perimeter of a square, width & length problems
- Real-world applications of relations and functions are presented with examples
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Graphs of Quadratic Functions: - Definition: function in the form y = ax2 + bx + c. - Graph: parabola. - Axis of symmetry: line of symmetry in quadratic graphs (x = −b/2a) - Vertex: maximum or minimum point on the parabola determined by finding x=-b/2a, and replacing the value found for x in the equation. - Vertex form: a(x - h)2 ± m.
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Solving Quadratic Equations Graphically:
- Finding the x-intercepts of a quadratic function.
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Solving Quadratic Inequalities Graphically: Graph the parabola and find the intervals of the x-values where the parabola falls.
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Description
Test your knowledge on the Cartesian coordinate system with this quiz. Explore the concepts of axes, quadrants, coordinate notation, and the properties of linear functions. Perfect for students learning about geometry and functions.
