Mathematics General - Graphs

Straight Line Graphs

  • Standard Form: \(y = mx + b\)

Features of a Straight Line Graph

  • X-intercept: substitute \(y=0\)
  • Y-intercept: value of \(b\)

Transformations of a Straight Line Graph

  • Vertical Translation Up: increase \(b\)
  • Vertical Translation Down: decrease \(b\)
  • Increase steepness: increase \(m\)
  • Decrease steepness: decrease \(m\)
  • Reflect in y-axis: \(m \times -1\)
  • Reflect in x-axis: \(y \times -1\) AND \(m \times -1\)
  • Reflect in Main Diagonal \((y=x)\): switch y and x
  • Horizontal Translation Left: increase \(b\)
  • Horizontal Translation Right: decrease \(b\)

Lines Parallel to the Axis

  • Standard Form (parallel to x-axis): \(y=b\)
  • Standard Form (Parallel to y-axis): \(x=a\)

Transformations of Lines Parallel to the Axis

  • Vertical translation up: Increase \(b\)
  • Vertical Translation Down: Decrease \(b\)
  • Horizontal Translation Left: Decrease \(a\)
  • Horizontal Translation Right: Increase \(a\)

Parabolas

  • General Form: \(y=ax^2 + bx + c\)

Features of a Parabola

  • X-Intercepts (not always present): intersects of parabola and \(y=0\)
  • Y-Intercept: intersects of parabola \(x=0\). ONLY 1 per Parabola
  • Axis of symmetry: vertical line \(\text{-}\) x of vertex
  • Vertex: turning point of Parabola
  • Minimum/Maximum Y value: y of vertex
  • Concavity: Does the graph face up or down?

Transformations of a Parabola

  • Dilating the graph: 1>x>0
  • Contracting the graph: a>1
  • Vertical Translation Up: increase the value of c
  • Vertical Translation Down: decrease the value of c
  • Horizontal Translation:\(\sqrt{c} \times -1\)
  • Reflect in X axis: \(a \times -1\)

Exponential Graphs

  • Standard Form: \(y=h^{x + n} + b\)

Features of an Exponential Graph

  • Y-Intercept - y-int=b+1
    • Proof:
      • Substitute x=0
      • y = h0 + b
      • h0 = 1
      • Therefore, y = 1+b
  • Asymptote: A line which the graph CANNOT cross, Asymptote = b
  • X-Intercept: Must be calculated manually, only present if b < 0

Transformations of an Exponential Graph

  • Move graph up: increase b
  • Move graph down: decrease b
  • Increase steepness: increase h
  • Decrease steepness: decrease h
  • Reflect in Y-axis: x * -1
  • Reflect in X-axis: h * -1
  • Move graph left n units: x + n
  • Move graph right n units: x - n

Circles

  • Standard Form: \((x-h)^2+(y-k)^2=r^2\)

Features of a Circle

  • Y-intercepts
  • X-intercepts
  • Note: if \(h\) and \(k\) are both zero, then the \(x\) and \(y\) intercepts are \(r\) and \(-1 \times r\)

Transformations of a Circle

Note: these will seem incorrect, but just try plotting them first. The confusion arises because the circle formula uses a minus sign.

  • Move graph up: decrease k
  • Move graph down: increase k
  • Increase steepness: increase x-coefficient
  • Decrease steepness: decrease x-coefficient
  • Increase width: increase y-coefficient
  • Decrease width: decrease y-coefficient
  • Move graph left: decrease h
  • Move graph right: increase h

Hyperbola

  • Standard Form: \(y = \frac{a}{x-h} +k\)

Transformations of a Hyperbola

  • Vertical Translation Up: increase k
  • Vertical Translation Down: decrease k
  • Horizontal Translation Left: decrease h
  • Horizontal Translation Right: increase h
  • Reflect in y-axis: \(a \times -1\)
  • Move vertices closer to center: decrease a
  • Move vertices further from center: increase a

Cubic Graph

  • Standard Form: \(y=(ax+b)^3+d\)

Transformations of a Cubic Graph

  • Vertical Translation Up: increase \(d\)
  • Vertical Translation Down: decrease \(d\)
  • Reflect in y-axis: \(a \times -1\)
  • Increase steepness: increase \(a\)
  • Decrease steepness: decrease \(a\)
  • Horizontal Translation Left: increase \(b\)
  • Horizontal Translation Right: decrease \(b\)


Post Author: [Pranav Sharma](mailto:rbxii3@rbxii3.com)
Pranav Sharma
Pranav Sharma
Site Owner

Year 12 Student, site owner and developer.

Related