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Graphing Quadratic Functions: ParabolasMATH 101 College Algebra
J. Robert Buchanan
Department of Mathematics
Summer 2012
J. Robert Buchanan Graphing Quadratic Functions: Parabolas
Objectives
In this lesson we will learn to:
graph a parabola (a quadratic function) and determine itsvertex, domain, range, line of symmetry, and zeros, and
solve applied problems by using quadratic functions andthe concept of maximum and minimum.
J. Robert Buchanan Graphing Quadratic Functions: Parabolas
Quadratic Functions
Definition
Any function that can be written in the form
y = ax2 + bx + c
where a, b, and c are real numbers with a 6= 0 is a quadraticfunction.
Remarks:
The graph of any quadratic function is a curve called aparabola.
Vertical parabolas are parabolas which “open upward” or“open downward”.
Horizontal parabolas are parabolas which “open left” or“open right”.
J. Robert Buchanan Graphing Quadratic Functions: Parabolas
Example
Consider the function y = x2 − 6x + 5. We can choose somevalues for x and compute the corresponding values for y .
J. Robert Buchanan Graphing Quadratic Functions: Parabolas
Example
Consider the function y = x2 − 6x + 5. We can choose somevalues for x and compute the corresponding values for y .
x y−1 12
0 51 02 −33 −44 −35 06 57 12
-2 2 4 6 8x
5
10
15
20
y
J. Robert Buchanan Graphing Quadratic Functions: Parabolas
Anatomy of a Parabola
Vertex: H3,-4L
x=3
-2 2 4 6 8x
-5
5
10
15
20
y
Observations:
The turning point at (3,−4) is called the vertex of theparabola.
The vertical line at x = 3 is called the line of symmetry ofthe parabola.
J. Robert Buchanan Graphing Quadratic Functions: Parabolas
Functions of the Form y = ax2
Consider the graphs of the following parabolas of the formy = ax2.
-3
-1
-
1
3
1
2
1
2
-4 -2 2 4x
-40
-20
20
40
y
J. Robert Buchanan Graphing Quadratic Functions: Parabolas
Remarks
For all quadratic functions, the domain is the set of all realnumbers.For parabolas of the form y = ax2,
the parabola opens upward if a > 0 and the range is y ≥ 0,the parabola opens downward if a < 0 and the range isy ≤ 0,the vertex is the point (0, 0),the line of symmetry is x = 0,the bigger |a| is, the narrower the opening of the parabola,andthe smaller |a| is, the wider the opening of the parabola.
J. Robert Buchanan Graphing Quadratic Functions: Parabolas
Functions of the Form y = ax2 + k
Consider the graphs of the following parabolas of the formy = ax2 + k (for these examples a = ±1).
-2
0
1
-2 -1 1 2x
-2
-1
1
2
3
4y
-2
01
-2 -1 1 2x
-4
-3
-2
-1
1
2y
J. Robert Buchanan Graphing Quadratic Functions: Parabolas
Remarks
Adding k to ax2 shifts or translates the graph verticallyby k units.For parabolas of the form y = ax2 + k ,
the parabola opens upward if a > 0 and the range is y ≥ k ,the parabola opens downward if a < 0 and the range isy ≤ k ,the vertex is the point (0, k),the line of symmetry is x = 0,the bigger |a| is, the narrower the opening of the parabola,andthe smaller |a| is, the wider the opening of the parabola.
J. Robert Buchanan Graphing Quadratic Functions: Parabolas
Functions of the Form y = a(x − h)2
Consider the graphs of the following parabolas of the formy = a(x − h)2 (for these examples a = ±1).
-10 -5 5 10x
50
100
150
200
y
4
0
-5
-10 -5 5 10x
-200
-150
-100
-50
y
-4
05
J. Robert Buchanan Graphing Quadratic Functions: Parabolas
Remarks
Subtracting h from x in a(x − h)2 shifts or translates thegraph horizontally by h units.
For parabolas of the form y = a(x − h)2,the parabola opens upward if a > 0 and the range is y ≥ 0,the parabola opens downward if a < 0 and the range isy ≤ 0,the vertex is the point (h, 0),the line of symmetry is x = h,the bigger |a| is, the narrower the opening of the parabola,andthe smaller |a| is, the wider the opening of the parabola.
J. Robert Buchanan Graphing Quadratic Functions: Parabolas
Functions of the Form y = ax2 + bx + c
y = ax2 + bx + c
= a(
x2 +ba
x)
+ c
= a(
x2 +ba
x +b2
4a2 −b2
4a2
)
+ c
= a(
x +b2a
)2
−b2
4a+ c
= a(x − h)2 + k
where
h = −b2a
k =4ac − b2
4a
J. Robert Buchanan Graphing Quadratic Functions: Parabolas
Functions of the Form y = ax2 + bx + c
y = ax2 + bx + c
= a(
x2 +ba
x)
+ c
= a(
x2 +ba
x +b2
4a2 −b2
4a2
)
+ c
= a(
x +b2a
)2
−b2
4a+ c
= a(x − h)2 + k
where
h = −b2a
k =4ac − b2
4aRemark: we can obtain the graph of y = ax2 + bx + c througha horizontal shift of h units and a vertical shift of k units.
J. Robert Buchanan Graphing Quadratic Functions: Parabolas
Examples
The following graphs show parabolas of y = a(x − h)2 + k fordifferent choices of h and k . The values of a used are ±1.
-4 -2 2 4x
20
40
60
80
y
H0,0L
H-3,0L
H-3,20L
H0,20L
-6 -4 -2 2 4 6x
-100
-80
-60
-40
-20
y
H0,-10L
H0,15L
H-4,-10LH-4,15L
J. Robert Buchanan Graphing Quadratic Functions: Parabolas
Remarks
Writing the standard form of a quadratic function
y = ax2 + bx + c = a(x − h)2 + k
where h = −b2a
and k =4ac − b2
4a
Subtracting h from x in a(x − h)2 shifts or translates thegraph horizontally by h units.Adding k to a(x − h)2 shifts or translates the graphvertically by k units.The parabola opens upward if a > 0 and the range isy ≥ k ,The parabola opens downward if a < 0 and the range isy ≤ k ,The vertex is the point (h, k), andThe line of symmetry is x = h.
J. Robert Buchanan Graphing Quadratic Functions: Parabolas
Zeros of a Quadratic Function
If the graph of the quadratic function y = ax2 + bx + c crossesthe x-axis, these points are called the zeros of the function.They can be found by solving the equation:
ax2 + bx + c = 0.
Example
Find the zeros of the function, the line of symmetry, thecoordinates of the vertex, the range, and the graph of theparabola for
y = x2 − 7x + 10
J. Robert Buchanan Graphing Quadratic Functions: Parabolas
Solution
Zeros of the function:
0 = x2 − 7x + 10 = (x − 2)(x − 5)
x = 2 or x = 5
J. Robert Buchanan Graphing Quadratic Functions: Parabolas
Solution
Zeros of the function:
0 = x2 − 7x + 10 = (x − 2)(x − 5)
x = 2 or x = 5
y = x2 − 7x + 10 = x2 − 7x +494
−494
+ 10
=
(
x −72
)2
−494
+ 10 =
(
x −72
)2
−94
Line of symmetry: x =72
Vertex:(
72
,−94
)
Domain: all real numbers (−∞,∞)
Range: y ≥ −94
since a = 1 > 0.
J. Robert Buchanan Graphing Quadratic Functions: Parabolas
Graph
Vertex: H7�2,-9�4L
x=7�2
2 4 6 8x
-5
5
10
15
20
y
J. Robert Buchanan Graphing Quadratic Functions: Parabolas
Maximum and Minimum Values
Observations: for parabolas which open upward, the vertex isthe lowest point on the parabola, while for parabolas whichopen downward, the vertex is the highest point on the parabola.
J. Robert Buchanan Graphing Quadratic Functions: Parabolas
Maximum and Minimum Values
Observations: for parabolas which open upward, the vertex isthe lowest point on the parabola, while for parabolas whichopen downward, the vertex is the highest point on the parabola.
Theorem
For a parabola with its equation in the form y = a(x − h)2 + k,1 if a > 0, the parabola opens upward and (h, k) is the
lowest point and y = k is called the minimum value of thefunction,
2 if a < 0, the parabola opens downward and (h, k) is thehighest point and y = k is called the maximum value ofthe function,
J. Robert Buchanan Graphing Quadratic Functions: Parabolas