How Do You Know When a Parabola Opens Up or Down
Parabolas
The graph of a quadratic equation in two variables (y = axtwo + bx + c ) is called a parabola. The post-obit graphs are two typical parabolas their ten-intercepts are marked past red dots, their y-intercepts are marked past a pinkish dot, and the vertex of each parabola is marked by a greenish dot:
We say that the first parabola opens upwards (is a U shape) and the second parabola opens downwards (is an upside down U shape). In gild to graph a parabola nosotros need to find its intercepts, vertex, and which way it opens.
Given y = ax2 + bx + c , we accept to get through the following steps to notice the points and shape of any parabola:
- Label a, b, and c.
- Decide the direction of the paraola:
- Observe the ten-intercepts:
- Detect the y-intercept:
- Find the vertex (h,k):
- Plot the points and graph the parabola
If a > 0 (positive) then the parabola opens upward.
If a < 0 (negative) then the parabola opens down.
Notice that the x-intercepts of any graph are points on the x-centrality and therefore have y-coordinate 0. We tin can observe these points by plugging 0 in for y and solving the resulting quadratic equation (0 = ax2 + bx + c). If the equation factors nosotros tin can find the points easily, just we may have to use the quadratic formula in some cases. If the solutions are imaginary, that means that the parabola has no x-intercepts (is strictly in a higher place or below the x-axis and never crosses information technology). If the solutions are real, but irrational (radicals) then we need to approximate their values and plot them.
The y-intercept of any graph is a point on the y-centrality and therefore has x-coordinate 0. We tin utilize this fact to find the y-intercepts past simply plugging 0 for ten in the original equation and simplifying. Notice that if nosotros plug in 0 for x we get: y = a(0)two + b(0) + c or y = c. So the y-intercept of any parabola is always at (0,c).
To observe the x-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:
k = a(h) 2 + b(h) + c
Case i) Graph y = xii + 2x - 8
In this problem: a = ane, b = 2 , and c = -eight.
Since "a" is positive nosotros'll have a parabola that opens upwards (is U shaped).
To discover the x-intercepts we plug in 0 for y:
0 = xii + 2x - viii (which factors)
0 = (x + 4)(x - 2)
x = -4 or x = two
So this parabola has two x-intercepts: (-4,0) and (2,0).
To detect the y-intercept nosotros plug in 0 for ten:
y = (0)2 + two(0) - 8 = -eight
And then the y-intercept of the parabola is (0,-eight).
To discover the vertex we use:
k = (-one)2 + 2(-1) - eight
k = 1 - 2 - eight = -9
The vertex of this parabola is at (-1, -9)
Example ii) Graph y = -3x2 + 3
In this problem a = -3, b = 0 and c = 3.
Since "a" is negative this parabola is going to open downward (upside down U shape).
To find the x-intercepts we plug in 0 for y:
0 = -3xii + 3 (this equation factors)
0 = -3(xtwo - 1)
0 = -3(10 - 1)(ten + 1) and since -3 can not equal zero:
ten = 1 or x = -1
The ten-intercepts are: (1,0) and (-ane,0)
The y-intercept is institute past plugging 0 for x:
y = -3(0)2 + iii = 3
Then, the y-intercept is at (0,3).
And to find the vertex:
yard = -3(0)2 + 3 = 3
Then the vertex is at (0,3).
Discover that in this problem the vertex and the y-intercept are the same bespeak.
Case 3) Graph y = x 2 + ivten + vii
a = i, b = 4, and c = 7
Since a 0 the parabola opens upwardly (is U shaped).
To find the x -intercept we plug in 0 for y:
0 = x 2 + 410 + 7 (this expression does not cistron and then nosotros have to utilise the quadratic formula)
Since the roots are imaginary the parabola has no x-intercepts.
We find the y-intercepts by plugging in 0 for x:
y = 02 + iv(0) + 7 = 7
The y-intercept is (0,vii).
The vertex:
Then the vertex is at (-2, 3).
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Source: http://www.csun.edu/~ayk38384/notes/mod11/Parabolas.html
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