IB MYP 3 Mathematics 5.2 Gradient and Straight-Line Graphs Study Notes - New Syllabus

5.2 – Gradient and Straight-Line Graphs

A straight-line graph represents a linear relationship between two variables. One of the most important features of a straight line is its gradient, which describes how steep the line is and how much the \(y\)-value changes when the \(x\)-value changes.

Gradient

The gradient of a straight line measures its rate of change. It tells us how much \(y\) changes for a given change in \(x\).

\(\text{Gradient}=\dfrac{\text{change in }y}{\text{change in }x}\)

The gradient is usually represented by the letter \(m\).

\(m=\dfrac{\Delta y}{\Delta x}\)

A positive gradient means that \(y\) increases as \(x\) increases. A negative gradient means that \(y\) decreases as \(x\) increases.

Finding the Gradient from Two Points

If two points on a straight line are:

\((x_1,y_1)\quad\text{and}\quad(x_2,y_2)\)

then the gradient is:

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

The order of the points does not matter as long as you use the same order in the numerator and denominator.

For example, using \(A(2,3)\) and \(B(6,11)\):

\(m=\dfrac{11-3}{6-2}\)

\(m=\dfrac{8}{4}=2\)

Therefore, the gradient is \(2\).

 Positive Gradient

A line has a positive gradient when it rises from left to right.

For example, consider the points \(A(1,2)\) and \(B(5,10)\).

\(m=\dfrac{10-2}{5-1}=\dfrac{8}{4}=2\)

Since \(m>0\), the line rises from left to right.

Negative Gradient

A line has a negative gradient when it falls from left to right.

For example, consider \(A(1,8)\) and \(B(5,2)\).

\(m=\dfrac{2-8}{5-1}=\dfrac{-6}{4}=-\dfrac{3}{2}\)

Since \(m<0\), the line falls from left to right.

Steepness of a Line

The absolute value of the gradient describes how steep a line is.

For example:

\(m=2\) has a greater steepness than:  \(m=\dfrac{1}{2}\)

Similarly, \(m=-4\) is steeper than \(m=-1\) because:

\(|-4|>|-1|\)

Gradient from a Graph

When a line is shown on a coordinate grid, you can calculate its gradient by choosing two points that lie exactly on the line.

Then:

\(m=\dfrac{\text{rise}}{\text{run}}\)

where:

• Rise = change in the \(y\)-direction.
• Run = change in the \(x\)-direction.

For example, if moving \(3\) units to the right causes the line to rise \(6\) units:

\(m=\dfrac{6}{3}=2\)

If moving \(4\) units to the right causes the line to fall \(3\) units:

\(m=\dfrac{-3}{4}\)

 Horizontal Lines

A horizontal line has no change in \(y\).

Therefore:

\(m=\dfrac{0}{\text{change in }x}=0\)

For example, the line passing through \((1,4)\) and \((7,4)\) has:

\(m=\dfrac{4-4}{7-1}=0\)

Its equation is:

\(y=4\)

 Vertical Lines

A vertical line has no change in \(x\). Its gradient would require division by zero:

\(m=\dfrac{\text{change in }y}{0}\)

Division by zero is undefined, so a vertical line has an undefined gradient.

For example, a vertical line through \(x=3\) has equation:

\(x=3\)

Straight-Line Graphs

A straight-line graph represents a relationship where the rate of change is constant.

This means that equal changes in \(x\) produce equal changes in \(y\).

For example, consider:

\(y=2x+1\)

The gradient is \(2\). Therefore, whenever \(x\) increases by \(1\), \(y\) increases by \(2\).

The points \((0,1)\), \((1,3)\), \((2,5)\) and \((3,7)\) all lie on the same straight line.

Gradient as Rate of Change

Gradient can also describe a real-world rate of change.

For example, suppose the distance travelled by a cyclist increases by \(15\) km every \(3\) hours.

\(m=\dfrac{15}{3}=5\)

The gradient is \(5\text{ km/h}\).

This means the distance increases at a constant rate of \(5\) km for every hour.

Comparing Gradients

When comparing two straight lines:

• A larger positive gradient means a steeper upward line.
• A smaller positive gradient means a less steep upward line.
• A more negative gradient means a steeper downward line.
• A gradient of \(0\) represents a horizontal line.

For example, compare:

\(m_1=3,\qquad m_2=\dfrac{1}{2}\)

Since:

\(3>\dfrac{1}{2}\)

the line with gradient \(3\) is steeper.