Lesson Notes By Weeks and Term v5 - Grade 10
Functions – Week 4 focus
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Functions – Week 4 focus
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Subject: Mathematics
Class: Grade 10
Term: 2nd Term
Week: 4
Theme: General lesson support
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This week, we delve deeper into the world of functions, building upon what we've learned so far. Functions are a fundamental building block in mathematics, acting as a mathematical "machine" that takes an input, processes it, and produces an output. Understanding functions allows us to model relationships between quantities and make predictions based on these relationships. For example, understanding how a cellular company charges per megabyte of data (a function relating data usage to cost) allows you to budget your data usage effectively. Similarly, understanding the function relating electricity consumption to cost helps manage your household expenses.
What is a Function? A function is a relationship between two sets (the domain and the range) where each element in the domain corresponds to exactly one element in the range. Think of it as a special kind of relationship. We can represent a function in several ways: Equation: A mathematical formula that defines the relationship.
Example: y = 2x + 1 Table: A table of values showing corresponding input (x) and output (y) values.
Graph: A visual representation of the relationship on a coordinate plane.
Domain and Range Domain: The set of all possible input values (x-values) for which the function is defined. In simpler terms, what numbers can you put into the function?
Range: The set of all possible output values (y-values) that the function can produce. What numbers come out of the function? Linear Functions A linear function has the general form: y = mx + c, where: m is the slope (gradient) of the line, indicating its steepness and direction. c is the y-intercept (the point where the line crosses the y-axis). To graph a linear function, you need at least two points. You can find these points by choosing two x-values, substituting them into the equation, and calculating the corresponding y-values.
Example 1: Graphing a Linear Function Graph the function y = x -
2. Determine the domain and range.
Solution: Choose two x-values: Let's choose x = 0 and x =
2. Calculate the corresponding y-values: When x = 0, y = 0 - 2 = -
2. So, we have the point (0, -2). When x = 2, y = 2 - 2 =
0. So, we have the point (2, 0). Plot the points (0, -2) and (2, 0) on a coordinate plane and draw a straight line through them.
Domain: Since x can be any real number, the domain is all real numbers, or (-∞, ∞).
Range: Similarly, y can be any real number, so the range is all real numbers, or (-∞, ∞). Quadratic Functions A quadratic function has the general form: y = ax 2 + bx + c, where a, b, and c are constants and a ≠
0. The graph of a quadratic function is a parabola.
Turning Point (Vertex): The minimum (if a > 0) or maximum (if a 2 - 4x +
3. Determine the domain and range.
Solution: Find the turning point: x-coordinate of the turning point: x = -b/(2a) = -(-4)/(21) = 2. y-coordinate of the turning point: y = (2) 2 - 4(2) + 3 = 4 - 8 + 3 = -
1. So, the turning point is (2, -1).
Find the y-intercept: Set x = 0: y = (0) 2 - 4(0) + 3 =
3. The y-intercept is (0, 3).
Find the x-intercepts: Set y = 0: 0 = x 2 - 4x +
3. Factor the quadratic: 0 = (x - 1)(x - 3). So, x = 1 or x =
3. The x-intercepts are (1, 0) and (3, 0). Plot the turning point, y-intercept, and x-intercepts and sketch the parabola.
Domain: Since x can be any real number, the domain is all real numbers, or (-∞, ∞).
Range: The parabola opens upwards (since a > 0), and the turning point is the minimum point.
Therefore, y ≥ -
1. The range is [-1, ∞). Exponential Functions An exponential function has the general form: y = a x , where a is a constant and a > 0 and a ≠
1. Asymptote: A line that the graph approaches but never touches. For the basic exponential function y = a x , the x-axis (y = 0) is a horizontal asymptote.
Intercept: The y-intercept is always (0, 1) for y = a x . If a > 1, the function is increasing (the graph rises as x increases). If 0 x . Determine the domain and range.
Solution: Create a table of values: | x | -2 | -1 | 0 | 1 | 2 | | :--- | :---- | :---- | :- | :- | :- | | y | 0.25 | 0.5 | 1 | 2 | 4 | Plot the points from the table and sketch the curve. Remember that the x-axis is an asymptote.
Domain: Since x can be any real number, the domain is all real numbers, or (-∞, ∞).
Range: The graph is always above the x-axis, so y >
0. The range is (0, ∞). Function Notation and Evaluation We often use function notation: f(x) instead of y. f(x) represents the output of the function f for the input x. To evaluate a function f(x) for a specific value of x, simply substitute that value into the function's equation.
Example 4: Function Evaluation If f(x) = 3x - 5, find f(2) and f(-1).
Solution: f(2) = 3(2) - 5 = 6 - 5 = 1. f(-1) = 3(-1) - 5 = -3 - 5 = -
8. Guided Practice (With Solutions)
Question 1: Represent the function described by the following table using an equation: | x | -2 | -1 | 0 | 1 | 2 | |---|---|---|---|---|---| | y | -5 | -3 | -1 | 1 | 3 | Solution: Notice that the y-values increase by 2 for every increase of 1 in the x-values. This suggests a linear function. The slope, m, is
2. The y-intercept is the y-value when x = 0, which is -
1. Therefore, the equation is y = 2x -
1. Question 2: Determine the domain and range of the function y = √( x - 3 ).
Solution: The square root function is only defined for non-negative values.
Therefore, x - 3 ≥ 0, which means x ≥
3. The domain is [3, ∞). Since the square root function always returns a non-negative value, y ≥
0. The range is [0, ∞).
Question 3: Sketch the graph of y = -x 2 + 2x +
3. Identify the turning point and intercepts.
Solution: Turning Point: x = -b/(2a) = -2/(2(-1)) = 1.