What information can the coefficients a, b, and c give you?
Axis of Symmetry: x=1 Minimum Value: - 4 Domain: All real numbers Range: y≥- 4
Practice makes perfect
We have a quadratic function written in standard form.
y= ax^2+ bx+ c
This kind of equation can give us a lot of information about the parabola by observing the values of a, b, and c.
y=4x^2-8x ⇔ y= 4x^2+( - 8)x+ 0
We see that for the given equation, a= 4, b= - 8, and c= 0.
We can identify the minimum or maximum value of a parabola by identifying the y-coordinate of its vertex. Whether the parabola has a minimum or maximum is determined by the value of a.
Since the given value of a is positive, the parabola has a minimum value at the vertex. To find this value, think of y as a function of x, y=f(x). Note that we already know the x-coordinate of the vertex since it lies on the axis of symmetry x= 1.
Vertex: ( 1, f( 1) )
By substituting the x-value of the vertex into the given equation and simplifying, we will get the y-value of the vertex, which is also the minimum value of the function.
The y-value of the vertex, therefore the maximum value of the function, is - 4.
Domain and Range
Unless there is a specific restriction given in the context of the problem, the domain of a quadratic function is all real numbers. In this case, there is no restriction on the value of x. Since the minimum value of the function is - 4, the range is all real numbers greater than or equal to - 4.
Range: y≥- 4
Extra
A Common Mistake
One common mistake when identifying the key features of a parabola algebraically is forgetting to include the negatives in the values of these constants. The standard form is addition only, so any subtraction must be treated as negative values of a, b, or c. Let's look at an example.
y=3x^2-4x-2 ⇕ y=3x^2 + (-4x) + (-2)
In this case, the values of a, b, and c are 3, -4, and -2. They are NOT 3, 4, and 2.
a=3, b=4, c=2 *
a=3, b=-4, c=-2 ✓
