Question

2. Determine the points of intersection of each pair of functions. a) y = 4x^– 15x + 20 and y = 5x – 4 = - - b) y = - 2x^ + 9x +9 and y = - 3x – 5

Answer 1

To determine the points of intersection we first equate the expressions, then we solve for x. Once we have the values of x for which the functions are equal we plu them on one of the function to find its corresponding value of y.

a)

Let's equate the functions and solve for x:

`\begin{gathered} 4x^2-15x+20=5x-4 \\ 4x^2-15x-5x+20+4=0 \\ 4x^2-20x+24=0 \\ 4(x^2-5x+6)=0 \\ x^2-5x+6=0 \\ (x-3)(x-2)=0 \\ \text{ then} \\ x=3 \\ or \\ x=2 \end{gathered}`

Now we find the corresponding values of y for each value of x; to do this we use the second equation.

When x=3:

`\begin{gathered} y=5(3)-4 \\ y=15-4 \\ y=11 \end{gathered}`

Hence the functions intersect at (3,11)

When x=2:

`\begin{gathered} y=5(2)-4 \\ y=10-4 \\ y=6 \end{gathered}`

Hence the functions intersect at (2,6)

Therefore the function intersect at the points (3,11) and (2,6).

b)

Let's equate the functions and solve for x:

`\begin{gathered} -2x^2+9x+9=-3x-5 \\ 2x^2-9x-9-3x-5=0 \\ 2x^2-12x-14=0 \\ 2(x^2-6x-7)=0 \\ x^2-6x-7=0 \\ (x-7)(x+1)=0 \\ \text{ then} \\ x=7 \\ or \\ x=-1 \end{gathered}`

Now we find the corresponding values of y for each value of x; to do this we use the second equation.

When x=7:

`\begin{gathered} y=-3(7)-5 \\ y=-21-5 \\ y=-26 \end{gathered}`

Hence the functions intersect at (7,-26)

When x=-1:

`\begin{gathered} y=-3(-1)-5 \\ y=3-5 \\ y=-2 \end{gathered}`

Hence the functions intersect at (-1,-2)

Therefore the function intersect at the points (7,-26) and (-1,-2).