a The parabola's function can be written in factored form as f(x)=a(x-b)(x-c), where b and c show the x-intercepts. The value of a decides if the parabola opens upwards or downwards.
B
b In this case, both x-intercepts are the same.
C
c Write the function in factored form.
D
d What causes the parabola to open up?
A
aExample Solution: f(x)=2(x+4)(x-2)
B
bExample Solution: f(x)=3(x-3)^2
C
cExample Solution: f(x)=3x(x-7)^2
D
dExample Solution: f(x)=-3(x+5)(x-7)
Practice makes perfect
a If we know the x-intercepts of a parabola we can write the function in factored form.
f(x)=a(x-b)(x-c)In this function b and c show the function's x-intercepts. By substituting b= - 4 and c= 2 in the factored form, we can write the parabola's function.
f(x)=a(x-( - 4))(x- 2)
⇓
f(x)=a(x+4)(x-2)
In addition to knowing the function's x-intercepts, we also know that the parabola opens up upwards. This is true if a>0. Whatever positive value we choose a to be, it will not change the x-intercept.
b Like in Part A, we can write the function in factored form.
f(x)=a(x-b)(x-c)In this case the function's roots is one and the same, x=3, which means b= c= 3. By substituting this value in the factored form we can write the parabola's function.
f(x)=a(x- 3)(x- 3)
⇓
f(x)=a(x-3)^2
Just like in Part A we know that the parabola opens up upwards, which means a is positive. We have no more information about the value of a, and therefore are free to choose whichever value we want.
c Like in Parts A and B, we can write the function in factored form.
f(x)=a(x-b)(x-c)We see that the parabola intercepts the x-axis at x=0 and x=7. Therefore, by substituting b= 0 and c= 7 in the factored form we can write the parabola's function.
f(x)=a(x- 0)(x- 7)
⇓
f(x)=ax(x-7)
Just like in Parts A and B, we know that the parabola opens up upwards, which means a is positive. We have no more information about the value of a, and therefore we are free to choose whichever value we want as long as it is positive.
d Like in previous parts, we can write the function in factored form.
f(x)=a(x-b)(x-c)We see that the parabola intercepts the x-axis at x=- 5 and x=1. Therefore, by substituting b= -5 and c= 1 in the factored form, we can write the parabola's function.
f(x)=a(x-( - 5))(x- 1)
⇓
f(x)=a(x+5)(x-1)
In this case, the parabola opens up downwards. Therefore, the value of a must be negative. We have no more information about the value of a, and therefore we are free to choose whichever value we want so long as it is negative.
