Some quadratic graphs are upwards, some downwards, some intersect with the x-axis, others not. What are the attributes of these graphs? How can we use them in graphing the function?
Attributes of a Quadratic Graph
The primary features of a quadratic graph are x and y-intercepts, vertex, and its orientation. These are essential for graphing the quadratic function. The coefficients of the quadratic function control these attributes. The following table lists these traits and their relation to the coefficients.
|Graph trait||Coefficients controlling the trait|
|Upward||A > 0|
|Downward||A < 0|
|Vertex||xv = − B/(2A), y = f(xv)|
|Axis of Symmetry||x = − B/(2A)|
|Y Intercept||Value of C|
|No x-intercept||D < 0|
|One x-intercept||D = 0|
|Two x-intercepts||D > 0|
|Graph Width||Magnitude of A|
Where A, B, C are coefficients of a quadratic function in the standard form:
f(x) = Ax2 + Bx + C and
D = B2 − 4AC
Before we graph a quadratic, let us understand these attributes.
Upward / Downward Graph
Coefficient A determines the shape of the quadratic graph and its orientation. It is upward or downward depending on the sign of the coefficient A.
y = x2 + 2x + 2
A (=1) is positive, the graph is upwards. Vertex is the lowest point (minimum value).
y = − x2 + 3x − 2
A (=-1) is negative, the graph is downwards. Vertex is the highest point (maximum value).
On the x-axis, as we move away from the origin (either towards –∞ or +∞):
- The value of x2 increases faster than all other terms combined
- x2 is always positive, so the sign of the term Ax2 is determined by A
- If A is positive, Ax2 goes to +∞ (on either direction of x-axis). Therefore the graph is upwards
- If A is negative, Ax2 goes to -∞ (on either direction of x-axis). Therefore the graph is downwards
Vertex of a Quadratic Function
Vertex is the turning point in a graph. There is only one vertex in a quadratic graph. It is symmetric across the vertical line passing through the vertex. This line is its axis of symmetry.
The following holds for the vertex:
- It is the highest ( or maximum ) point when A < 0 (downward graph)
- It is the lowest ( or minimum ) point when A > 0 (upward graph)
- x-coordinate = − B/(2A)
- Axis of symmetry passes through it
Find the vertex of a quadratic function y = x2 + 2x + 1. Using the formula above, you can calculate x and y values of the vertex as:
- x = − B/(2A) = − 2/2 = − 1
- y = f(x) = f( − 1) = 1 − 2 + 1 = 0
The axis of symmetry passes through x = -1 and is parallel to the y-axis.
The y-intercept is a point where the graph hits the y-axis. The value of x is zero on the y-axis. So its y-coordinate is easy to calculate:
y = f(0) = A.02 + B.0 + C = C
Therefore y-intercept is always equal to (0, C).
The value of C increases (or decreases, if C is negative) the value of the function.
If C is positive, the graph shifts up, and if C is negative, it shifts down.
The graphs below show this shift using the following functions:
- y-intercept = ( 0, -1 )
- y-intercept = ( 0 , 1 )
- y-intercept = ( 0, 2 )
Comparing y-values of y-intercept
The difference in y-values of g(x) and f(x) is 2 ( = 1 – (-1) ). Similarly, the difference in y-values of h(x) and g(x) is 1 ( = 2 – 1 ). The graphs are shifted vertically (by the difference of respective C values). On the y-axis, value of function equals the corresponding value of C.
As C is the y-intercept, even the graphs of different quadratic functions with the same value of C will coincide on the y-axis.
The x-intercept is a point where a graph intersects the x-axis. You would have observed above that not all the graphs intersect the x-axis. If it does, it has a root (or roots). The x-intercept is the root.
The existence of roots depends on the sign of the discriminant D.
- D = B2 − 4AC
- If D > 0, there are two x-intercepts
- If D = 0, graph just touches the x-axis and therefore only one x-intercept
- If D < 0, the graph will be above or below the x-axis. There are no x-intercepts
✩ Formula – Roots of a Quadratic Function
x1 = ( − B + √D)/(2A)
x2 = ( − B − √D)/(2A)
where D = B2 − 4AC
Example: No x-intercept
When the discriminant is negative, the graph will not touch the x-axis. There is no x-intercept. You can see this for the function below:
- f(x) = x2 + 2x + 2
- Discriminant of function f(x) = B2 − 4AC = − 4
- x-intercept: none
Example: One x-intercept
If the discriminant is zero, the graph will touch the x-axis at one point, so there is only one x-intercept.
- Graph of x2 + 2x + 1
- Discriminant = 0
- x-intercept: ( − 1, 0)
Example: Two x-intercepts
In the graph below, the corresponding function has a positive discriminant. The graph intersects the x-axis at two points. It has two x-intercepts.
- Graph of
- Discriminant = 25/4( > 0)
- x-intercepts: ( 0.5, 0 ) and ( 3, 0 )
Width of Graph
The width of a graph is a comparative feature. Between any two quadratic graphs, the one with a larger magnitude of A will be narrower.
The x2 term grows faster than all the other terms combined. Coefficient A has a multiplying effect on the growth of x2. Its impact depends on |A| (magnitude of A). So the function with the highest |A| grows ( or falls, for A < 0 ) fastest. Its graph is the narrowest.
For example, if we have the following two functions:
- f(x) = 2x2 + 1 |A| = 2
- g(x) = − 5x2 + 1 |A| = 5
As |-5| > |2|, g(x) has a narrower graph.
The graphs below correspond to the following functions:
- g(x) = 2x2 + 1
|A| is highest for the function g(x). So its graph is the narrowest and grows fastest. The value of |A| for f(x) and h(x) are the same. Graphs of both the functions have the same width.
Graphing Quadratic Function in the Standard Form
Using the above traits, we can quickly graph the quadratic function. We will use the following steps:
- Determine if the graph is upwards or downwards
- Find its vertex
- Calculate x and y-intercepts
- Finally, using the axis of symmetry, find a few mirrored points and plot the graph.
Let us see an example.
Graph the function f(x) = x2 − 3x + 2
Coefficients of function f(x): A = 1, B = -3, C = 2
As coefficient A > 0, we have an upward graph.
Let us find the vertex using the formula:
x = − B/(2A) = 3/2 = 1.5
y = f(x) = f(1.5)
= (1.5)2 − 3(1.5) + 2
= 4.25 − 4.5 = − 0.25
The coordinates of the vertex are ( 1.5, -0.25 )
Coordinates of y-intercept are: (0, C) = (0, 2)
Let us find the coordinates of the x-intercepts. We have to calculate the discriminant first:
D = B2 − 4AC
D = ( − 3)2 − 4(1)(2) = 9 − 8 = 1
As discriminant D > 0, there are two x-intercepts ( roots ). Using the following formula for roots of a quadratic function, we calculate the x-intercepts.
x1 = (3 + 1)/2x2 = (3 − 1)/2
x1 = 4/2 = 2x2 = 2/2 = 1
The x-intercepts are: ( 2, 0 ) and ( 1, 0 ).
We calculate a few more points to make our graph accurate. We use the ones at an equal horizontal distance from the axis to reduce our effort by half.
|x||y = f(x)|
|-1.0||f(-1.0) = 6.0|
|-0.5||f(-0.5) = 3.75|
|0.0||f(0) = 2.0|
|0.5||f(0.5) = 0.75|
|2.5||f(2.5) = 0.75|
|3.0||f(3.0) = 2.0|
|3.5||f(3.5) = 3.75|
|4.0||f(4.0) = 6.0|
We now plot vertex ( 1.5 , -0.25 ), y-intercept ( 0, 2 ), x-intercepts ( 1, 0 ) & ( 2, 0 ) along with the points above.