We have a written in .
y=ax^2+ bx+c
This kind of equation can give us a lot of information about the by observing the values of a, b, and c.
y=- x^2+6x+5 ⇔ y= - 1x^2+ 6x+ 5
We see that for the given equation, a= - 1, b= 6, and c= 5.
Axis of Symmetry
The is the that passes through the , dividing a parabola into two mirror images.
Note that its equation follows a specific formula.
x=- b/2 a
We can now substitute the given values of a and b into the expression and simplify.
The axis of symmetry is the line x=3.
Maximum or Minimum Value
We can identify the minimum or maximum value of a parabola by identifying the y-coordinate of its vertex. The value of a tells us whether the parabola has a minimum or a maximum.
Since the given value of a is negative, the parabola has a maximum 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= 3.
Vertex: ( 3, f( 3) )
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 maximum value of the function.
y=- x^2+6x+5
y=- ( 3)^2+6( 3)+5
y=14
The y-value of the vertex, therefore the maximum value of the function, is 14.
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 maximum value of the function is 14, the range is all real numbers less than or equal to 14.
Range: y≤ 14
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 ✓
