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 \(b\)