Straight-Line Graphs

Straight-Line Graphs

Every straight line in the coordinate plane has an equation of the form \( y = mx + c \), where \( m \) is the gradient (steepness and direction) and \( c \) is the y-intercept (the value of \( y \) where the line crosses the y-axis). Reading, drawing and finding the equations of straight lines are fundamental skills used throughout algebra and the analysis of real-world data.

Gradient and y-intercept

\[ y = mx + c qquad m = \text{gradient} = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1} \]

A positive \( m \) gives an upward slope; negative \( m \) slopes downward. Steeper lines have larger \( |m| \).

Finding the Equation from Two Points

  1. Calculate the gradient: \( m = \dfrac{y_2 - y_1}{x_2 - x_1} \).
  2. Substitute one point and \( m \) into \( y = mx + c \) to find \( c \).
  3. Write the full equation.

Parallel Lines

Parallel lines have equal gradients and different \( y \)-intercepts. They never intersect.

−2 0 2 4 6 x 5 4 3 2 1 −1 y y = ½x + 2 y = ½x − 1 Both lines: m = ½ (parallel)

Worked Examples

Plot \( y = 2x - 3 \). State the gradient and y-intercept.

Gradient \( m = 2 \) (rises 2 for every 1 step right). y-intercept \( c = -3 \) (crosses y-axis at \( (0,-3) \)).

Plot two points: at \( x = 0 \), \( y = -3 \); at \( x = 3 \), \( y = 3 \). Draw a straight line through them.

Find the equation of the line through \( (1, 5) \) and \( (3, 11) \). Then through \( (2, 1) \) with gradient \( -\frac{1}{2} \).

Two points: \( m = \frac{11-5}{3-1} = 3 \). Substitute \( (1,5) \): \( 5 = 3(1)+c \Rightarrow c = 2 \). Equation: \( y = 3x+2 \).

One point + gradient: \( y - 1 = -\frac{1}{2}(x-2) \Rightarrow y = -\frac{1}{2}x + 2 \).

Find the equation of the line parallel to \( y = 3x - 2 \) passing through \( (2, 7) \).

Parallel lines share the same gradient: \( m = 3 \). Use the given point: \( 7 = 3(2) + c \Rightarrow c = 1 \).

Equation: \( y = 3x + 1 \).

 Key Takeaways

  • In \( y = mx + c \): \( m \) is the gradient (change in \( y \) per unit change in \( x \)); \( c \) is where the line crosses the \( y \)-axis.
  • Gradient between two points: \( m = \dfrac{y_2 - y_1}{x_2 - x_1} \). Maintain consistent order of subtraction in numerator and denominator.
  • Equation through one point with gradient \( m \): use \( y - y_1 = m(x - x_1) \), then rearrange to \( y = mx + c \).
  • Parallel lines have equal gradients. They are distinct if their \( c \) values differ; identical if \( c \) is also the same.