Graphs and Transformations
Reciprocal Graphs
Courtesy of Lumen Learning
Points of Intersection
● Coordinates of POI satisfy the equations of both curves/lines
● Solve equations simultaneously to get coordinates of POI
● Mark on graph when sketching
Proportional Relationships
● If y is directly proportional to x then the relationship between y and x can be
expressed as y ∝ x or y = kx where k is a constant
● Graphs of proportional relationships are straight lines through the origin
● If y is inversely proportional to x then the relationship between y and x can
be expressed as y ∝ 1x or y = kx where k is a constant
● This means that as one quantity increases, the other decreases
Graph Translations
y = f (x) + a
● This is a translation of the graph y = f (x) by a units parallel to the y-axis
y = f (x − a)
● This is a translation of the graph y = f (x) by a units parallel to the x-axis
y = f (x − a) + b
● This is a translation of the graph y = f (x) by a units parallel to the x-axis
and b units parallel to the y-axis
Reciprocal Graphs
Courtesy of Lumen Learning
Points of Intersection
● Coordinates of POI satisfy the equations of both curves/lines
● Solve equations simultaneously to get coordinates of POI
● Mark on graph when sketching
Proportional Relationships
● If y is directly proportional to x then the relationship between y and x can be
expressed as y ∝ x or y = kx where k is a constant
● Graphs of proportional relationships are straight lines through the origin
● If y is inversely proportional to x then the relationship between y and x can
be expressed as y ∝ 1x or y = kx where k is a constant
● This means that as one quantity increases, the other decreases
Graph Translations
y = f (x) + a
● This is a translation of the graph y = f (x) by a units parallel to the y-axis
y = f (x − a)
● This is a translation of the graph y = f (x) by a units parallel to the x-axis
y = f (x − a) + b
● This is a translation of the graph y = f (x) by a units parallel to the x-axis
and b units parallel to the y-axis
