How to graph a quadratic functions :
Here we are going to see some example problems on graphing quadratic functions.
A quadratic function can be described by an equation of the form y = ax^{2} + bx + c, where a ≠ 0.
Open upward parabola |
Open downward parabola |
(1) (i) If the coefficient of x^{2} is positive, the parabola opens upward.
(ii) If the coefficient of x^{2} is negative, the parabola opens upward.
This point, where the parabola changes direction, is called the "vertex".
We can find the x-coordinate of the vertex by using the formula, x = -b/2a
By applying the value of x in the given equation, we can get the y-coordinate value.
Vertex of the parabola (-b/2a, f(-b/2a)).
By giving some random values of x, we can find the values of y.We can use the symmetry of the parabola to help us draw its graph. On a coordinate plane, graph the vertex and the axis of symmetry.
Let us look into some example problems to understand the above concept.
Example 1 :
Write the equation of the axis of symmetry, and find the coordinates of the vertex of the graph of each function. Identify the vertex as a maximum or minimum. Then graph the function.
y = 2x^{2} - 4x - 5
Solution :
Axis of symmetry :
The given parabola is symmetric about y-axis. Since the coefficient of x^{2} is positive, the parabola opens upward direction.
Vertex of the parabola :
x = -b/2a
a = 2, b = -4 and c = -5
x = -(-4)/2(2) ==> 4/4 ==> 1
By applying the value x = 1 in the given equation
y = 2(1)^{2 } - 4(1) - 5
= 2 - 4 - 5
= -7
Hence the vertex of the parabola is (1, -7).
Identify the vertex as a maximum or minimum :
Since the parabola opens upward direction, it has only minimum value.
Draw the graph :
x -2 -1 0 1 2 |
y 11 1 -5 -7 -5 |
Set of ordered pairs : (-2, 11) (-1, 1) (0, -5) (1, -7) and (2, -5) |
Example 2 :
Write the equation of the axis of symmetry, and find the coordinates of the vertex of the graph of each function. Identify the vertex as a maximum or minimum. Then graph the function.
y = -3x^{2} - 6x + 4
Solution :
Axis of symmetry :
The given parabola is symmetric about y-axis. Since the coefficient of x^{2} is negative, the parabola opens downward direction.
Vertex of the parabola :
x = -b/2a
a = -3, b = -6 and c = 4
x = -(-6)/2(-3) ==> 6/(-6) ==> -1
By applying the value x = -1 in the given equation
y = -3(-1)^{2} - 6(-1) + 4
= -3 + 6 + 4
= -3 + 10
= 7
Hence the vertex of the parabola is (-1, 7).
Identify the vertex as a maximum or minimum :
Since the parabola opens downward direction, it has only maximum value.
Draw the graph :
x -2 -1 0 1 2 |
y -4 7 4 -5 -20 |
Set of ordered pairs : (-2, -4) (-1, 7) (0, 4) (1, -5) and (2, -20) |
After having gone through the stuff given above, we hope that the students would have understood "How to graph a quadratic functions".
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