\( \DeclareMathOperator{cosec}{cosec} \)

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International Baccalaureate Mathematics

Functions

Syllabus Content

Different forms of the equation of a straight line.
Gradient, intercepts.
Parallel lines m1 = m2.
Perpendicular lines m1 × m2 = -1.

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Furthermore

Official Guidance, clarification and syllabus links:

\(y=mx+c\) (gradient-intercept form).

\(ax+by+d=0\) (general form).

\(y − y_1=m(x − x_1)\) (point-gradient form).

Calculate gradients of inclines such as mountain roads, bridges, etc.

Formula Booklet:

Equations of a straight line

\( y=mx+c; \\ ax+by+d=0; \\ y - y_1 = m(x-x_1) \)

Gradient formula

\( m = \dfrac{y_2 - y_1}{x_2 - x_1} \)

The equation of a straight line can be expressed in various forms, each highlighting different aspects of the line's geometric properties. The most common forms are:

1. Slope-Intercept Form: \( y = mx + c \), where \( m \) is the gradient (slope) and \( c \) is the y-intercept. This form is useful for easily identifying the slope and y-intercept of the line.

$$ y = mx + c $$

Example: The line with gradient 3 and y-intercept -2 is given by \( y = 3x - 2 \).

2. Point-Slope Form: \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a point on the line and \( m \) is the gradient. This form is helpful when a point on the line and the slope are known.

$$ y - y_1 = m(x - x_1) $$

Example: A line passing through the point (2, 3) with a gradient of 4 is given by \( y - 3 = 4(x - 2) \).

3. General Form: \( Ax + By + C = 0 \), where A, B, and C are constants. This form is useful for certain algebraic manipulations and geometric interpretations.

$$ Ax + By + C = 0 $$

Example: The line with equation \( 2x + 3y - 6 = 0 \) can be rearranged to slope-intercept form as \( y = -\frac{2}{3}x + 2 \).

For parallel and perpendicular lines:

Parallel Lines: Two lines are parallel if they have the same gradient. For lines \( y = m_1x + c_1 \) and \( y = m_2x + c_2 \), they are parallel if \( m_1 = m_2 \).

Perpendicular Lines: Two lines are perpendicular if the product of their gradients is -1. For lines \( y = m_1x + c_1 \) and \( y = m_2x + c_2 \), they are perpendicular if \( m_1 \times m_2 = -1 \).

$$ \text{For parallel lines: } m_1 = m_2 $$ $$ \text{For perpendicular lines: } m_1 \times m_2 = -1 $$

This video on Forms of Linear Lines is from Revision Village and is aimed at students taking the IB Maths AI Standard level course

This video on Gradients and Intercepts of Linear Lines is from Revision Village and is aimed at students taking the IB Maths AI Standard level course.

This video on Parallel and Perpendicular Gradients is from Revision Village and is aimed at students taking the IB Maths Standard level course

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