how to find the x intercepts of a parabola

Parabolas

The graph of a quadratic equation in 2 variables (y = ax²+ bx + c) is chosen a parabola. The following graphs are two typical parabolas their x-intercepts are marked by cerise dots, their y-intercepts are marked by a pink dot, and the vertex of each parabola is marked by a green dot:

We say that the starting time parabola opens upwards (is a U shape) and the second parabola opens downwards (is an upside down U shape). In order to graph a parabola we demand to detect its intercepts, vertex, and which mode it opens.

Given y = ax² + bx + c , we have to become through the following steps to find the points and shape of any parabola:

Label a, b, and c.

Decide the direction of the paraola:

If a > 0 (positive) then the parabola opens upwards.
If a < 0 (negative) then the parabola opens down.

Find the 10-intercepts:

Notice that the ten-intercepts of any graph are points on the ten-axis and therefore have y-coordinate 0. We tin detect these points by plugging 0 in for y and solving the resulting quadratic equation (0 = ax² + bx + c). If the equation factors nosotros can notice the points easily, but we may take to use the quadratic formula in some cases. If the solutions are imaginary, that ways that the parabola has no x-intercepts (is strictly above or below the 10-centrality and never crosses it). If the solutions are real, but irrational (radicals) and so nosotros need to approximate their values and plot them.

Detect the y-intercept:

The y-intercept of any graph is a point on the y-axis and therefore has x-coordinate 0. We tin use this fact to find the y-intercepts past simply plugging 0 for x in the original equation and simplifying. Notice that if we plug in 0 for ten nosotros get: y = a(0)² + b(0) + c or y = c. So the y-intercept of any parabola is always at (0,c).

Find the vertex (h,m):

To find the ten-coordinate for the vertex nosotros use the following formula:

To find the y-coordinate for the vertex we plug in h in the original equation:
1000 = a(h) ⁱⁱ + b(h) + c

Plot the points and graph the parabola

Example ane) Graph y = x² + 2x - 8

In this problem: a = 1, b = ii , and c = -8.

Since "a" is positive nosotros'll have a parabola that opens up (is U shaped).
To observe the ten-intercepts we plug in 0 for y:
0 = x² + 2x - viii (which factors)
0 = (x + 4)(x - 2)
10 = -iv or ten = 2
So this parabola has 2 x-intercepts: (-4,0) and (2,0).

To find the y-intercept we plug in 0 for x:
y = (0)² + 2(0) - viii = -viii
So the y-intercept of the parabola is (0,-8).

To find the vertex we use:

and to find k, we plug in -1 in for 10:
k = (-1)ⁱⁱ + 2(-1) - 8
g = 1 - 2 - 8 = -ix
The vertex of this parabola is at (-1, -9)

Example ii) Graph y = -3x^two + 3

In this problem a = -3, b = 0 and c = 3.

Since "a" is negative this parabola is going to open up downwards (upside downward U shape).

To find the ten-intercepts we plug in 0 for y:
0 = -3x² + 3 (this equation factors)
0 = -3(ten² - 1)
0 = -iii(x - 1)(ten + 1) and since -3 can not equal zippo:
x = 1 or x = -i
The x-intercepts are: (1,0) and (-1,0)

The y-intercept is establish by plugging 0 for ten:
y = -three(0)² + 3 = iii
Then, the y-intercept is at (0,three).

And to find the vertex:

chiliad = -three(0)² + 3 = three
And then the vertex is at (0,3).

Notice that in this problem the vertex and the y-intercept are the same point.

Example 3) Graph y = ten ² + fourx + 7

a = 1, b = 4, and c = 7

Since a 0 the parabola opens upwardly (is U shaped).

To notice the 10 -intercept we plug in 0 for y:
0 = x ² + 4x + 7 (this expression does not gene so we have to use the quadratic formula)

Since the roots are imaginary the parabola has no ten-intercepts.