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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 a and q for the parabola of the form y=ax 2 +q.

Image

Answer

Examine the sketch

From the sketch we see that the shape of the graph is a “frown”, therefore a<0. We also see that the graph has been shifted vertically upwards, therefore q>0.

Determine q using the y-intercept

The y-intercept is the point (0;1).

y=ax 2 +q1=a(0) 2 +qq=1(1)

Use the other given point to determine a

Substitute point (-1;0) into the equation:

y=ax 2 +q0=a(-1) 2 +1a=-1(2)

Write the final answer

a=-1 and q=1, so the equation of the parabola is y=-x 2 +1.

Example 2: Determining the equation of a hyperbola

Question

Use the sketch below to determine the values of a and q for the hyperbola of the form y=a x+q.

Image

Answer

Examine the sketch

The two curves of the hyperbola lie in the second and fourth quadrant, therefore a<0. We also see that the graph has been shifted vertically upwards, therefore q>0.

Substitute the given points into the equation and solve

Substitute the point (-1;2):

y=a x+q2=a -1+q2=-a+q(3)

Substitute the point (1;0):

y=a x+q0=a 1+qa=-q(4)

Solve the equations simultaneously using substitution

2=-a+q=q+q=2qq=1a=-q=-1(5)

Write the final answer

a=-1 and q=1, the equation of the hyperbola is y=-1 x+1.

Example 3: Interpreting graphs

Question

The graphs of y=-x 2 +4 and y=x-2 are given. Calculate the following:

  1. coordinates of A, B, C, D

  2. coordinates of E

  3. distance CD

Image

Answer

Calculate the intercepts

For the parabola, to calculate the y-intercept, let x=0:

y=-x 2 +4=-0 2 +4=4(6)

This gives the point C(0;4).

To calculate the x-intercept, let y=0:

y=-x 2 +40=-x 2 +4x 2 -4=0(x+2)(x-2)=0x=±2(7)

This gives the points A(-2;0) and B(2;0).

For the straight line, to calculate the y-intercept, let x=0:

y=x-2=0-2=-2(8)

This gives the point D(0;-2).

For the straight line, to calculate the x-intercept, let y=0:

y=x-20=x-2x=2(9)

This gives the point B(2;0).

Calculate the point of intersection E

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

x-2=-x 2 +4x 2 +x-6=0(x-2)(x+3)=0x=2or-3(10)

At E, x=-3, therefore y=x-2=-3-2=-5. This gives the point E(-3;-5).

Calculate distance CD

CD=CO+OD=4+2=6(11)

Distance CD is 6 units.

Example 4: Interpreting trigonometric graphs

Question

Use the sketch to determine the equation of the trigonometric function f of the form y=af(θ)+q.

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 y=asinθ+q.

Substitute the given points into equation and solve

At N, θ=210 and y=0:

y=asinθ+q0=asin210 +q=a-1 2+qq=a 2(12)

At M, θ=90 and y=3 2:

3 2=asin90 +q=a+q(13)

Solve the equations simultaneously using substitution

3 2=a+q=a+a 23=2a+a3a=3a=1q=a 2=1 2(14)

Write the final answer

y=sinθ+1 2(15)