d Note that we can factor g(x) by pulling out an x from the binomial.
A
a (0,-6)
B
b x=-3 and x=2
C
c g(x)=x^2+x
D
d x-intercepts: x=0 and x=-1
y-intercept: (0,0) Diagram:
Practice makes perfect
a For any function, the value of its constant tells us where it intersects the y-axis. If we examine the equation, we can identify the constant.
f(x)=x^2+x - 6
This function has a constant of -6, which means it intersects the y-axis at (0,-6).
b To determine where the graph intersects the x-axis, we will factor the right-hand side. Let's use a generic rectangle and a diamond problem. We know that x^2 and -6 go into the lower-left and upper-right corners of the generic rectangle, respectively.
Now we To fill in the remaining two corners, we need two x-terms that have a sum of x and a product of -6x^2.
Notice that the product is negative. This means one factor must be positive and the other must be negative.
Product
ax(bx)
ax+bx
Sum
x?
- 6x^2
- 3x(2x)
- 3x+2x
- x
*
- 6x^2
- 2x(3x)
- 2x+3x
x
✓
When one factor is -2x and the other is 3x, we get a product of -6x^2 and a sum of x. Now we can complete the diamond and generic rectangle.
To factor the right-hand side, we add each side of the generic rectangle and multiply the sums.
f(x)=x^2+x-6
⇕
f(x)=(x-2)(x+3)
Now we will we set f(x) equal to 0 and then use the Zero Product Property to solve the equation and find the x-intercepts.
d As previously noted, the y-intercept of a function is given by the constant. If a function has no constant, it means that the y-intercept is the origin. To determine the x-intercepts, we will set g(x) equal to 0, factor the right-hand side, and solve for x using the Zero Product Property.
