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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 $y=ax2+q$.

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 $(0;1)$.

$y=ax2+q1=a(0)2+q∴q=1$(1)

Use the other given point to determine a

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

$y=ax2+q0=a(-1)2+1∴a=-1$(2)

$a=-1$ and $q=1$, so the equation of the parabola is $y=-x2+1$.

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 $y=ax+q$.

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 $(-1;2)$:

$y=ax+q2=a-1+q∴2=-a+q$(3)

Substitute the point $(1;0)$:

$y=ax+q0=a1+q∴a=-q$(4)

Solve the equations simultaneously using substitution

$2=-a+q=q+q=2q∴q=1∴a=-q=-1$(5)

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

Example 3: Interpreting graphs

Question

The graphs of $y=-x2+4$ and $y=x-2$ are given. Calculate the following:

1. coordinates of $A$, $B$, $C$, $D$

2. coordinates of $E$

3. distance $CD$

Calculate the intercepts

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

$y=-x2+4=-02+4=4$(6)

This gives the point $C(0;4)$.

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

$y=-x2+40=-x2+4x2-4=0(x+2)(x-2)=0∴x=±2$(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:

$x-2=-x2+4∴x2+x-6=0∴(x-2)(x+3)=0∴x=2or-3$(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 $y=af(θ)+q$.

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 $N$, $θ=210ࢪ$ and $y=0$:

$y=asinθ+q0=asin210ࢪ+q=a-12+q∴q=a2$(12)

At $M$, $θ=90ࢪ$ and $y=32$:

$32=asin90ࢪ+q=a+q$(13)

Solve the equations simultaneously using substitution

$32=a+q=a+a23=2a+a3a=3∴a=1∴q=a2=12$(14)

$y=sinθ+12$(15)