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Mathematics General - Graphs

Last updated on January 5, 2022 3 min read Year 10, Mathematics

Straight Line Graphs

Standard Form: $y = m x + b$

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: $y = a x 2 + b x + c$

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 $-$ 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: $y = a x - h + k$

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: $y = (a x + b) 3 + d$

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 $b$

Graphs Year 10 Mathematics