Interpretation of graphs
Example 1: Determining the equation of a parabola
Answer
Examine the sketch
From the sketch we see that the shape of the graph is a “frown”, therefore . We also see that the graph has been shifted vertically upwards, therefore .
Determine using the intercept
The intercept is the point .
Use the other given point to determine a
Substitute point into the equation:
Write the final answer
and , so the equation of the parabola is .
Example 2: Determining the equation of a hyperbola
Answer
Examine the sketch
The two curves of the hyperbola lie in the second and fourth quadrant, therefore . We also see that the graph has been shifted vertically upwards, therefore .
Substitute the given points into the equation and solve
Substitute the point :
Substitute the point :
Solve the equations simultaneously using substitution
Write the final answer
and , the equation of the hyperbola is .
Example 3: Interpreting graphs
Question
The graphs of and are given. Calculate the following:

coordinates of , , ,

coordinates of

distance
Answer
Calculate the intercepts
For the parabola, to calculate the intercept, let :
This gives the point .
To calculate the intercept, let :
This gives the points and .
For the straight line, to calculate the intercept, let :
This gives the point .
For the straight line, to calculate the intercept, let :
This gives the point .
Calculate the point of intersection
At the two graphs intersect so we can equate the two expressions:
At , , therefore . This gives the point .
Calculate distance
Distance is 6 units.
Example 4: Interpreting trigonometric graphs
Answer
Examine the sketch
From the sketch we see that the graph is a sine graph that has been shifted vertically upwards. The general form of the equation is .
Substitute the given points into equation and solve
At , and :
At , and :