To find equation from the graph of parabola, first we known about what is vertex.
If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function.
If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value.
The vertex form of a quadratic polynomial is
y = a(x - h)2 + k
Here (h, k) is vertex.
Write the equation of the parabola in vertex form.
Problem 1 :
Solution :
The parabola opens up. The minimum point is at (0, -5).
Vertex (h, k) ==> (0, -5)
Equation of parabola :
y = a(x - h)2 + k
y = a(x - 0)2 + (-5) ----(1)
By observing the graph, the parabola passes through the point (2, -1).
-1 = a(2 - 0)2 + (-5)
-1 = 4a + (-5)
-1 + 5 = 4a
4a = 4
a = 4/4
a = 1
By applying the value of a in (1), we get
y = 1(x - 0)2 + (-5)
y = x2 - 5
Problem 2 :
Solution :
The parabola opens up. The minimum point is at (0, 2).
Vertex (h, k) ==> (0, 2)
Equation of parabola :
y = a(x - h)2 + k
y = a(x - 0)2 + 2 ----(1)
By observing the graph, the parabola passes through the point (1,3).
3 = a(1 - 0)2 + 2
3 = a + 2
a = 3 - 2
a = 1
By applying the value of a in (1), we get
y = 1(x - 0)2 + 2
y = x2 + 2
Problem 3 :
Solution :
The parabola opens up. The minimum point is at (2, 0).
Vertex (h, k) ==> (2, 0)
Equation of parabola :
y = a(x - h)2 + k
y = a(x - 2)2 + 0 ----(1)
By observing the graph, the parabola passes through the point (3, 1).
1 = a(3 - 2)2
1 = a(1)
a = 1
By applying the value of a in (1), we get
y = 1(x - 2)2
y = (x - 2)2
Problem 4 :
Solution :
The parabola opens up. The maximum point is at (0, 4).
Vertex (h, k) ==> (0, 4)
Equation of parabola :
y = a(x - h)2 + k
y = a(x - 0)2 + 4 ----(1)
By observing the graph, the parabola passes through the point (1, 2).
2 = a(2 - 0)2 + 4
2 = a(4) + 4
4a = 2 - 4
4a = -2
a = -2/4
a = -1/2
By applying the value of a in (1), we get
y = (-1/2)(x - 0)2 + 4
y = (-1/2)x2 + 4
Problem 5 :
Solution :
The parabola opens up. The maximum point is at (-3, 5).
Vertex (h, k) ==> (-3, 5)
Equation of parabola :
y = a(x - h)2 + k
y = a(x - (-3))2 + 5
y = a(x + 3)2 + 5----(1)
By observing the graph, the parabola passes through the point (-4, 2).
2 = a(-4 + 3)2 + 5
2 = a(1)2 + 5
a = 2 - 5
a = -3
By applying the value of a in (1), we get
y = -3(x + 3)2 + 5
May 21, 24 08:51 PM
May 21, 24 08:51 AM
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