Sketching Exponential Graphs

Edexcel IAL WMA12/01/P2/June 2021/Q2 (Sketching​​ Exponential Graphs, Logarithmic​​ Equations)

Figure 1 shows a sketch of the curve with equation y = 4x​​ 

A copy of Figure 1, labelled Diagram 1, is shown on​​ the next page.​​ 

• On Diagram 1, sketch the curve with equation​​ 

• y = 2x​​ 

• y = 4x​​ – 6​​ 

Label clearly the coordinates of any points of intersection with the coordinate axes.​​ 

(4)

The curve with equation y = 2x meets the curve with equation y = 4x​​ – 6 at the point P.​​ 

• Using algebra, find the exact coordinates of P.​​ 

(4)

SOLUTION

a- i- ​​ The curve must be less steeper than y = 4x.​​ 

a-ii- ​​ The curve of​​ y = 4x​​ is translated 6 units downwards along the y-axis. The y-intercept of​​ y = 4x​​ was 1; therefore the new y-intercept is​​ 1-6=-5

b- ​​ Since P is the point of intersection of the curves, so solving the equations simultaneously.

y = 2x

y = 4x-6

Equating them both​​ 

2x=4x-6

2x=22x-6

2x=2x2-6

let  a=2x, hence the euqation becomes​​ 

a=a2-6

a2-a-6=0

Solving quadratic equation by breaking the middle term method.​​ 

a+2a-3=0

a=-2,       a=3

a=-2​​ is rejected because the argument of log can neither be zero nor negative.​​ 

a=3

Now, substituting the value of ‘a’ back (  a=2x).

2x=3

x=log2⁡3

Now, finding the y-cordinate​​ 

y = 2x

Where we found above that​​ 2x=3, so​​ 

y = 2x=3

y=3

Hence, the coordinate the point P are​​ (log2⁡3,3)