The graphical form of a quadratic function will be a parabola. The quadratic equation can be converted into different forms.
The quadratic equation which is in the form of
y = ax2 + bx + c
will form a parabola which opens up or opens down.
The quadratic equation which is in the form of
y = a(x - h)2 + k
Here (h, k) is the vertex.
The factored form of a parabola with zeroes r and s.
y = a (x - r) (x - s)
Problem 1 :
Find a quadratic model for each set of values.
(–1, 1), (1, 1), (3, 9)
Solution :
Let the quadratic model be y = ax2 + bx + c
The parabola passes through the given points, so we can apply the points one by one.
It passes through (-1, 1).
1 = a(-1)2 + b(-1) + c
1 = a - b + c
a - b + c = 1 -----------(1)
It passes through (1, 1).
1 = a(1)2 + b(1) + c
1 = a + b + c
a + b + c = 1 -----------(2)
It passes through (3, 9).
9 = a(3)2 + b(3) + c
9 = 9a + 3b + c
9a + 3b + c = 9 -----------(3)
(1) + (2)
a - b + c + a + b + c = 1 + 1
2a + 2c = 2
a + c = 1 ----(4)
Multiply the first equation and subtract (3), we get
3a - 3b + 3c + 9a + 3b + c = 3+9
12a + 4c = 12 ----(5)
4(4) - (5)
4a + 4c - 12a - 4c = 4 - 12
-8a = -8
a = 1
Applying a = 1 in (4), we get 1 + c = 1 c = 0 |
When c = 0 and a = 1 1 + b + 0 = 1 b = 0 |
So, the required equation of parabola is y = x2
Problem 2 :
Find the quadratic function with the given vertex and point.
Vertex (2, 0) passing through (1, 3).
Solution :
y = a(x - h)2 + k
(h, k) ==> (2, 0)
y = a(x - 2)2 + 0
y = a(x - 2)2 ----(1)
The parabola is passing through the point (1, 3).
3 = a(1 - 2)2
3 = a(-1)2
3 = a
Applying the value of a in (1), we get
y = 3(x - 2)2
Problem 3 :
Determine the equation of the parabola whose zeroes are 4 and -5 which passes through (2, -42).
Solution :
Zeroes are 4 and -5.
x = 4 and x = -5
Product of factors :
(x - 4)(x + 5)
Writing it as factored form, we get
y = a (x - 4)(x + 5) ------(1)
The parabola passes through the point (2, -42).
-42 = a (2 - 4)(2 + 5)
-42 = a (-2)(7)
a = 42/14
a = 3
By applying the value of a in (1), we get
y = 3(x - 4)(x + 5)
We can convert this into standard form or vertex form. We can check the answer using graphing calculator.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM