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Interpretation of graphs

Example 1: Determining the equation of a parabola

Question

Use the sketch below to determine the values of and for the parabola of the form .

Image

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 .

(1)

Use the other given point to determine a

Substitute point into the equation:

(2)

Write the final answer

and , so the equation of the parabola is .

Example 2: Determining the equation of a hyperbola

Question

Use the sketch below to determine the values of and for the hyperbola of the form .

Image

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 :

(3)

Substitute the point :

(4)

Solve the equations simultaneously using substitution

(5)

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:

  1. coordinates of , , ,

  2. coordinates of

  3. distance

Image

Answer

Calculate the intercepts

For the parabola, to calculate the -intercept, let :

(6)

This gives the point .

To calculate the -intercept, let :

(7)

This gives the points and .

For the straight line, to calculate the -intercept, let :

(8)

This gives the point .

For the straight line, to calculate the -intercept, let :

(9)

This gives the point .

Calculate the point of intersection

At the two graphs intersect so we can equate the two expressions:

(10)

At , , therefore . This gives the point .

Calculate distance

(11)

Distance is 6 units.

Example 4: Interpreting trigonometric graphs

Question

Use the sketch to determine the equation of the trigonometric function of the form .

Image

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 :

(12)

At , and :

(13)

Solve the equations simultaneously using substitution

(14)

Write the final answer

(15)
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