IB Mathematics AA SL The graph of linear equation function Study Notes
IB Mathematics AA SL The graph of linear equation function Study Notes Offer a clear explanation of The graph of linear equation function , including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic The graph of linear equation function
The Graph of Linear Equation Function
A linear equation graph represents a straight line that shows the relationship between two variables, typically written in the form \( y = mx + c \).
The formula for a linear equation is:
\( y = mx + c \)
Where:
- \( y \) = the dependent variable.
- \( x \) = the independent variable.
- \( m \) = the slope of the line, showing the rate of change of \( y \) with respect to \( x \).
- \( c \) = the y-intercept, indicating where the line crosses the y-axis.
For example, let’s graph the equation \( y = 2x + 3 \):
- The slope \( m = 2 \) means the line rises by 2 units for every 1 unit it moves to the right.
- The y-intercept \( c = 3 \) means the line crosses the y-axis at (0, 3).
To plot this line:
- Start at the y-intercept (0, 3).
- Use the slope to find another point: from (0, 3), move up 2 units and 1 unit to the right to reach (1, 5).
- Draw a straight line through these points.
Properties of Linear Equations
Linear equations have several essential properties that define their characteristics:
1. Constant Rate of Change: The slope of a linear equation is constant, meaning the change in \( y \) with respect to \( x \) remains the same across the line.
2. Graph as a Straight Line: A linear equation represents a straight line when graphed on a coordinate plane, with no curves or bends.
3. Intercepts: The y-intercept is the point where the line crosses the y-axis (at \( x = 0 \)), and the x-intercept is the point where it crosses the x-axis (at \( y = 0 \)).
4. Equation Form: Linear equations can be written in the form \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.
5. Proportional Relationships (when \( c = 0 \)): If the y-intercept \( c \) is zero, the linear equation represents a proportional relationship where \( y \) is directly proportional to \( x \).
IB Mathematics AA SL The graph of linear equation function Exam Style Worked Out Questions
Question
Point P has coordinates (-3 , 2), and point Q has coordinates (15, -8). Point M is the midpoint of [PQ].
(a) Find the coordinates of M.
Line L is perpendicular to [PQ] and passes through M.
(b) Find the gradient of L.
(c) Hence, write down the equation of L.
▶️Answer/Explanation
Answer:
(a) M (6, -3)
(b) gradient of [PQ] = -\(\frac{5}{9}\)
gradient of L = \(\frac{5}{9}\)
(c) y + 3 = \(\frac{9}{5}\)(x – 6) OR y = \(\frac{9}{5}\) x – \(\frac{69}{5}\) (or equivalent)
Question
Consider the points A (-2 , 20) , B (4 , 6) and C (-14 , 12) . The line L passes through the point A and is perpendicular to [BC] .
(a) Find the equation of L . [3]
The line L passes through the point (k , 2) .
(b) Find the value of k .
▶️Answer/Explanation
Ans
b.
substituting (k , 2) into their L
2 − 20 = 3(k + 2)
OR
2 = 3k + 26
k=-8