Lesson Notes By Weeks and Term v5 - Grade 9
Functions and graphs (linear and simple non-linear) – Week 8 focus
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Functions and graphs (linear and simple non-linear) – Week 8 focus
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Subject: Mathematics
Class: Grade 9
Term: 2nd Term
Week: 8
Theme: General lesson support
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Functions and graphs are fundamental tools in mathematics that describe relationships between quantities. This week, we will explore linear functions and introduce simple non-linear functions, focusing on their graphical representation and algebraic manipulation. Understanding these concepts is crucial because they form the basis for more advanced mathematics and are applied in various real-world situations, from calculating cellphone data costs to modelling population growth and predicting trends in business.
2. 1. Linear Functions A linear function is a function whose graph is a straight line. The general form of a linear equation is y = mx + c, where: y is the dependent variable (usually plotted on the vertical axis). x is the independent variable (usually plotted on the horizontal axis). m is the slope or gradient, representing the rate of change of y with respect to x. A positive slope indicates an increasing line, a negative slope indicates a decreasing line, and a slope of zero indicates a horizontal line. It can be calculated using two points (x₁, y₁) and (x₂, y₂) on the line using the formula: m = (y₂ - y₁) / (x₂ - x₁)* c is the y-intercept, which is the point where the line crosses the y-axis (i.e., the value of y when x = 0).
Example 1: Consider the linear function y = 2x +
3. Slope (m) =
2. This means for every 1 unit increase in x, y increases by 2 units. The line is increasing. y-intercept (c) =
3. This means the line crosses the y-axis at the point (0, 3). To graph this function, we can find two points. Let's find the x-intercept (where y=0): 0 = 2x + 3 -3 = 2x x = -1.5 So, we have the points (0,3) and (-1.5, 0). Plot these points and draw a straight line through them.
Example 2: A cellphone company charges R1 per minute (slope) plus a fixed call connection fee of R2 (y-intercept). Write the equation representing the total cost (y) for x minutes of call time.
Solution: y = 1x + 2 or y = x + 2. 2.
2. Finding the Equation of a Line Given two points (x₁, y₁) and (x₂, y₂), we can find the equation of the line as follows: Calculate the slope: m = (y₂ - y₁) / (x₂ - x₁) Use the point-slope form of a linear equation: y - y₁ = m(x - x₁) Rearrange the equation into the slope-intercept form: y = mx + c Example 3: Find the equation of the line passing through the points (1, 5) and (3, 9).
Slope: m = (9 - 5) / (3 - 1) = 4 / 2 = 2 Using point-slope form with point (1, 5): y - 5 = 2(x - 1)
Rearrange: y - 5 = 2x - 2 => y = 2x + 3 2.
3. Simple Non-Linear Functions: Parabolas A parabola is a U-shaped curve. We will focus on parabolas of the form y = ax² + c, where: a determines the direction and "width" of the parabola. If a > 0, the parabola opens upwards (minimum point). If a y = x + 3 Therefore, the equation of the line is y = x +
3. Question 3: Sketch the graph of the parabola y = 2x² -
8. Identify the vertex and the axis of symmetry.
Solution: a = 2 (positive), so the parabola opens upwards. c = -8, so the y-intercept is (0, -8), and the vertex is (0, -8). The axis of symmetry is x =
0. If x = 1, y = 2(1)² - 8 = -
6. Point (1, -6). Symmetric point (-1,-6). If x = 2, y = 2(2)² - 8 =
0. Point (2,0). Symmetric point (-2, 0). Plot these points and draw a smooth U-shaped curve. Independent Practice (Questions Only) Sketch the graph of y = 3x -
1. Find the equation of the line passing through (0, -2) and (3, 4). A taxi service charges R10 as a call-out fee and R5 per kilometer. Write an equation to represent the total cost (y) of a taxi ride for x kilometers. Sketch the graph of y = -x² +
4. Identify the vertex and axis of symmetry. Determine the equation of a line with a slope of -2 and passing through the point (1, 3). Compare the graphs of y = x and y = 2x. How does the slope affect the steepness of the line? The cost of airtime is R2 per minute. Represent this relationship graphically, showing the cost for up to 10 minutes of call time. Describe the differences between the graph of y = x² and y = -x². Find the equation of the line passing through the points (4, 2) and (6, 8). If a parabola has a vertex at (0, 5) and opens downward, what can you say about the value of 'a' in its equation y = ax² + c?