# 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 $$b$$

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