The quadratic function which is in the standard form will be
y = ax^{2} + bx + c
this will have two roots α and β.
Roots = zeroes = x-intercepts
To find quadratic function from the given roots, we follow the steps give below.
Step 1 :
From the given graph, mark the zeroes and write them as values of x and convert it into factored form.
x = α ==> (x - α) (factor)
x = β ==> (x - β) (factor)
y = a(x - α)(x - β)
Step 2 :
To figure out the value of a, we will get one more point from the graph given except zeroes.
(i) It may be maximum or minimum point or y-intercept.
(ii) Mark it as point and apply in the quadratic function
(iii) Figure the value of a and simplify.
Determine the equation of quadratic function from graph. Give the function in general form.
Example 1 :
Solution :
From the graph, x-intercepts are 1 and 4.
So, x = 1 and x = 4
Factored form :
y = a(x - 1)(x - 4)
By observing the graph, the parabola cuts the y-axis at -4. By writing it as point (0, -4).
-4 = a(0 - 1)(0 - 4)
-4 = a(-1)(-4)
-4 = 4a
a = -1
y = -1(x - 1)(x - 4)
y = -1(x^{2} - 5x + 4)
y = -x^{2} + 5x - 4
Here the coefficient of x^{2} is -1, from the graph we have evidence that the parabola opens down.
Example 2 :
Solution :
From the graph, x-intercepts are 1 and 4.
So, x = 1 and x = 4
Factored form :
y = a(x - 1)(x - 4)
By observing the graph, the parabola cuts the y-axis at -4. By writing it as point (0, 4).
4 = a(0 - 1)(0 - 4)
4 = a(-1)(-4)
4 = 4a
a = 1
y = 1(x - 1)(x - 4)
y = 1(x^{2} - 5x + 4)
y = x^{2} - 5x + 4
Here the coefficient of x^{2} is 1, from the graph we have evidence that the parabola opens up.
Example 3 :
Solution :
From the graph, x-intercepts are -4 and 1.
So, x = -4 and x = 1
Factored form :
y = a(x + 4)(x - 1)
By observing the graph, the parabola cuts the y-axis at 12. By writing it as point (0, 12).
12 = a(0 + 4)(0 - 1)
12 = -4a
a = -3
y = -3(x + 4)(x - 1)
y = -3(x^{2} + 3x - 4)
y = -3x^{2} - 9x + 12
Here the coefficient of x^{2} is -3, from the graph we have evidence that the parabola opens down.
Example 4 :
Solution :
From the graph, x-intercepts are -4 and 1.
So, x = -4 and x = 1
Factored form :
y = a(x + 4)(x - 1)
By observing the graph, the parabola cuts the y-axis at -8. By writing it as point (0, -8).
-8 = a(0 + 4)(0 - 1)
-8 = -4a
a = 2
y = 2(x + 4)(x - 1)
y = 2(x^{2} + 3x - 4)
y = 2x^{2} + 6x - 8
Here the coefficient of x^{2} is 2, from the graph we have evidence that the parabola opens up.
Example 5 :
Solution :
From the graph, x-intercepts are 4 and 1.
So, x = 4 and x = 1
Factored form :
y = a(x - 4)(x - 1)
By observing the graph, the parabola cuts the y-axis at -8. By writing it as point (0, 8).
8 = a(0 - 4)(0 - 1)
8 = 4a
a = 2
y = 2(x - 4)(x - 1)
y = 2(x^{2} - 5x + 4)
y = 2x^{2} - 10x + 8
Here the coefficient of x^{2} is 2, from the graph we have evidence that the parabola opens up.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
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