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: $y=a{x}^2+bx+c$

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 $b$