A quadratic function is one with the form f(x) = ax2 + bx + c, where a, b, and c are all positive numbers, with a not being zero.
It is a curve called a parabola that represents the graph of a quadratic function. There are many types of parabolas, which open upward or downward and vary in “width” or “steepness”, but they all have the same basic “U” shape. Here is a picture which illustrates three graphs, and all of them are parabolas.
As a general rule, parabolas are all symmetric with respect to a line called the axis of symmetry. A parabola has a vertex, which is the point at which the axis of symmetry intersects the parabola’s axis of symmetry.
A line is formed by the intersection of two points. In other words, if you are given any two points in the plane, then there is always one and only one line that contains both points. It is also possible to make a similar statement with regard to quadratic functions.
There is exactly one quadratic function f whose graph contains all three points given three points in the plane that have different first coordinates and do not lie on a line. This is illustrated in the applet below. The graph contains three points and a parabola that passes through them all. The text underneath the graph shows the function that corresponds with each point. The function and parabola are updated when you drag any of the points.
A number of quadratic functions can be graphed easily by hand using the techniques of stretching/shrinking and shifting (translation) the parabola y = x2
Example No 1
Draw the graph of y = x2/2. The graph of y = x2 shrinks by half when we start at y = x2. In other words, for every point on the graph of y = x2, we draw a new point halfway from the x-axis to that point.
Example No 2
Sketch the graph of y = (x – 4) ^2 – 5. We start with the graph of y = x2, shift 4 units right, then 5 units down.
Standard Form of Quadratic Function
In Exercise 1, parts (a) and (b) depict quadratic functions represented in standard form. The standard form of a quadratic function makes it easy to draw the graph of the function by reflecting, shifting and stretching/shrinking the parabola y = x2.
The quadratic function f(x) = a(x – h)2 + k, with a not equal to zero, is said to be in standard form. In the case of a positive value, the graph opens upward, and in the case of a negative value, it opens downward. A line of symmetry is defined by the horizontal line x = h, and the vertex is defined as the point (h, k).
It is possible to rewrite any quadratic function in standard form by completing the square. If you have not already done so, please refer to the section on solving equations algebraically in order to review completing the square. The steps that we use in this section for completing the square will look a little different since our main objective here is not just to solve an equation.
It is also important to remember that when a quadratic function is in standard form, it is also easy to find its zeros by following the principle of square roots.
Example No 1
The function f(x) = x2 – 6x + 7 should be written in standard form. Draw the graph of f and determine its zeros and vertex.
- f(x) = x2 – 6x + 7
= (x2 – 6x) + 7 => Group the x2 and x terms and then complete the square on these terms.
= (x2 – 6x + 9 – 9) + 7
It is necessary to add 9 because it is one-half of (-6)2/2, which equals 9. Adding 9 to both sides of an equation is how we solved an equation. This setting adds and subtracts 9 in order to not change the function.
= (x2 – 6x + 9) – 9 + 7. We see that x2 – 6x + 9 is a perfect square, namely (x – 3)2.
Equation in the Standard Form – As a Result
- f(x) = (x – 3)2 – 2
From this result, one easily finds the vertex of the graph of f is (3, -2).
To find the zeros off, we set f equal to 0 and solve for x.
(x – 3)2 – 2 = 0.
(x – 3)2 = 2.
(x – 3) = ± sqrt(2)
x = 3 ± sqrt(2)
In order to sketch the graph of f, we shift the graph of y = x2 three units to the right and two units down.
If x2 does not have a coefficient of 1, then we must factor it from x2 and x terms before proceeding.