Question

How many intercepts does `y = x^2 + x − 6` have?

Answer 1

This function has a total of three intercepts.

Explanation:

There are two types of intercepts a function can have, `y`-intercepts, which are calculated for `x = 0`, and `x`-intercepts, which are calculated for `y = 0`.

So, starting with the `y`-intercept, you have `x = 0`, which results in

`y = (0)^2 + 0 - 6`

`y = -6`

This means that the `y`-intercept, which is the point where the graph of the function intercepts the `y`-axis, will be `(0,-6)`.

Now for the `x`-intercept. In order to find the `x`-intercept, you need to have `y=0`, which means that

`y = x^2 + x - 6 = 0`

The discriminant of this quadratic equation will actually tell you how many `x`-intercepts you have.

For ageneral form quadratic equation

`color(blue)(ax^2 + bx + c = 0)`

the discriminant is defined as

`color(blue)(Delta = b^2 - 4ac)`

For your quadratic, `a = 1`, `b=1`, and `c = -6`. The discriminant will be equal to

`Delta = 1^2 - 4 * 1 * (-6)`

`Delta = 1 + 24 = 25`

When `Delta>0`, the quadratic equation will have two disctinct real roots, which is another way of saying that the graph of the function will intercept the `x`-axis in two points.

These roots will be equal to

`color(blue)(x_(1,2) = (-b +- sqrt(Delta))/(2a)`

In your case, you have

`x_(1,2) = (-1 +- sqrt(25))/(2 * 1)`

`x_(1,2) = (-1 +- 5)/2 = {(x_1 = (-1 - 5)/2 = -3), (x_2 = (-1 + 5)/2 = 2):}`

The two `x`-intercepts will thus be `(-3,0)` and `(2, 0)`.

Therefore, the function will have a total of three intercepts, one `y`-intercept and two `x`-intercepts.

graph{x^2 + x - 6 [-20.27, 20.28, -10.14, 10.13]}