Linear graphs

Tables of Values

To draw \(y = 2x + 1\), work out \(y\) for a few values of \(x\), plot the points and join them with a ruler.

• Substitute

  When \(x = -2\), \(y = 2 \times (-2) + 1 = -3\).

• Three points at least

  Two points fix a line; a third catches a mistake.

• Straight line

  If the points are not in a straight line, check your table.

• Across the grid

  Draw the line through all the points, from edge to edge of the given range.

A Table of Values for y = 2x + 1

• \(-2\)

  Working: \(2 \times (-2) + 1\). y: \(-3\)

• \(-1\)

  Working: \(2 \times (-1) + 1\). y: \(-1\)

• 0

  Working: \(2 \times 0 + 1\). y: 1

• 1

  Working: \(2 \times 1 + 1\). y: 3

• 2

  Working: \(2 \times 2 + 1\). y: 5

• 3

  Working: \(2 \times 3 + 1\). y: 7

Horizontal and Vertical Lines

Some lines have only one letter in their equation.

• \(y = b\)

  A horizontal line: every point on it has \(y\)-coordinate \(b\). The \(x\)-axis is \(y = 0\).

• \(x = a\)

  A vertical line: every point on it has \(x\)-coordinate \(a\). The \(y\)-axis is \(x = 0\).

• \(y = x\)

  A diagonal line through the origin, where both coordinates are equal.

• \(y = -x\)

  The other diagonal through the origin.

Gradient

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

• Positive

  The line goes up from left to right.

• Negative

  The line goes down from left to right.

• Zero

  A horizontal line has gradient 0.

• From two points

  Between \((x_1, y_1)\) and \((x_2, y_2)\), gradient \(= \dfrac{y_2 - y_1}{x_2 - x_1}\).

y = mx + c

When the equation is written as \(y = mx + c\), you can read the line straight off it.

• \(m\)

  The gradient - the number in front of \(x\).

• \(c\)

  The \(y\)-intercept: the line crosses the \(y\)-axis at \((0, c)\).

• Rearrange first

  \(y\) must be on its own: \(2y = 4x + 6\) becomes \(y = 2x + 3\).

• Order doesn't matter

  \(y = 5 - 3x\) has gradient \(-3\) and \(y\)-intercept 5.