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5.1 Quadratic Functions
Chapter 5 Polynomial and Rational Functions
Concepts and Objectives
⚫ Objectives for this section are
⚫ Recognize characteristics of parabolas.
⚫ Understand how the graph of a parabola is related to
its quadratic function.
⚫ Determine a quadratic function’s minimum or
maximum value.
⚫ Solve problems involving a quadratic function’s
minimum or maximum value.
Quadratic Functions
⚫ A function f is a quadratic function if
where a, b, and c are real numbers, and a  0.
⚫ This is called the general form of the quadratic function.
⚫ The graph of a quadratic function is a parabola whose
shape and position are determined by a, b, and c.
⚫ If a > 0, the parabola opens upward.
⚫ If a < 0, the parabola opens downward.
⚫ |a| determines the width of the parabola.
( )= + +
2
f x ax bx c
Vertex Formula
⚫ We can use the general form to find the equation for the
axis of symmetry and the vertex of the parabola:
⚫ If we use the quadratic formula, to
solve for the zeros, we find the value of x
halfway between them is always the equation
for the axis of symmetry.
⚫ To find the y-coordinate of the vertex, plug x into the
general form.
2
4
,
2
b b ac
x
a
−  −
=
2
0
ax bx c
+ + =
,
2
b
x
a
= −
Vertex Form
⚫ The vertex (or standard) form of a quadratic function is
written
⚫ The graph of this function is the same as that of g(x)
translated h units horizontally and k units vertically.
This means that the vertex of f is at (h, k) and the axis of
symmetry is x = h.
⚫ The vertex form is just a vertical and horizontal shift of
the parent function y = x2.
( ) ( )
2
f x a x h k
= − +
Vertex Form (cont.)
⚫ Example: Graph the function and give its domain and
range.
( ) ( )
2
1
4 3
2
f x x
= − − +
Vertex Form (cont.)
⚫ Example: Graph the function and give its domain and
range.
Compare to : h = 4 and k = 3 (Notice
the signs!)
Vertex: (4, 3), axis of symmetry x = 4
We can graph this function by graphing the base
function and then shifting it.
( ) ( )
2
1
4 3
2
f x x
= − − +
( ) ( )
2
k
f x a x h
= − +
Vertex Form (cont.)
⚫ Example, cont.:
Let’s consider the graph of
⚫ Vertex is at (0, 0)
⚫ Passes through (2, ‒2) and
(4, ‒8).
⚫ (I picked 2 and 4 because of
the half.)
( )= − 2
1
2
g x x
Vertex Form (cont.)
⚫ Example, cont.:
To graph f, we just shift everything over 4 units to the
right and 3 units up.
Domain: (‒, )
Range: (‒, 3]
Writing a Function From a Graph
⚫ Given a graph of a quadratic function, to write the
equation of the function in general form:
⚫ Identify the vertex, (h, k). It will always be either the
highest or lowest point of the parabola.
⚫ Substitute the values of h and k in the function
⚫ Substitute the values of any point, other than the
vertex, on the parabola for x and f(x), and solve for a.
⚫ Expand and simplify to write in general form.
( ) ( )
2
f x a x h k
= − +
Writing a Function From a Graph
⚫ Example: Write an equation for the quadratic function g
in vertex form, and then in general form.
Writing a Function From a Graph
⚫ Example: Write an equation for the quadratic function g
in vertex form, and then in general form.
First, we identify the vertex,
which is at (–2, –3). Thus,
h = –2 and k= –3, and our
function looks like this:
( ) ( )
( ) ( )
( ) ( )
2
2
2 3
2 3
g x a x
g x a x
= − + −
= + −
Writing a Function From a Graph
⚫ Example: Write an equation for the quadratic function g
in vertex form, and then in general form.
Now, we pick a point on the
graph, for example, (0, –1),
to substitute in for x and
g(x).
( )
2
1 0 2 3
4 2
1
2
a
a
a
− = + −
=
=
Writing a Function From a Graph
⚫ Example: Write an equation for the quadratic function g
in vertex form, and then in general form.
So, the vertex form for this
graph is
( ) ( )
2
1
2 3
2
g x x
= + −
Writing a Function From a Graph
⚫ Example: Write an equation for the quadratic function g
in vertex form, and then in general form.
To write this in general
form, expand the binomial
and simplify:
( ) ( )
( )
2
2
2
2
1
2 3
2
1
4 4 3
2
1
2 2 3
2
1
2 1
2
g x x
x x
x x
x x
= + −
= + + −
= + + −
= + −
General Form to Vertex Form
⚫ Given a quadratic function in general form, to find the
vertex of the parabola (or to write in vertex form):
⚫ Identify a, b, and c from the general form.
⚫ Find h, the x-coordinate of the vertex, by substituting
a and b into .
⚫ Find k, the y-coordinate of the vertex, by evaluating
k .
⚫ Write in vertex form:
2
b
h
a
= −
( )
2
b
f h f
a
 
= = −
 
 
( ) ( )
2
f x a x h k
= − +
General Form to Vertex Form
⚫ Example: What is the vertex of the function?
( )= − +
2
6 7
f x x x
General Form to Vertex Form
⚫ Example: What is the vertex of the function?
From the general form, a = 1, b = –6, and c = 7.
The vertex is at (3, ‒2).
Vertex form is
( )= − +
2
6 7
f x x x
( )
6
3
2 1
h
−
= − = ( ) ( )
2
3 3 6 3 7
9 18 7 2
k f
= = − +
= − + = −
( ) ( )
2
3 2
f x x
= − −
Completing the Square
⚫ Another method we can use is “completing the square”
to transform it into vertex form (see section 2.5), but this
time we do everything on the right-hand side.
⚫ For example, the function is not a
binomial square. We can add 0 in the form of 52 – 52
(5 is half of 10), and group the parts that factor to a
binomial square:
( ) + −
= − +
2
2 2
10 30
5 5
f x x x
( )= − +
2
10 30
f x x x
( )
= − + − +
2 2 2
10 5 5 30
x x
( )
= − +
2
5 5
x
Completing the Square (cont.)
⚫ Example: What is the vertex of the function? (same ex.)
( )= − +
2
6 7
f x x x
Completing the Square (cont.)
⚫ Example: What is the vertex of the function?
The vertex is at (3, ‒2).
( )= − +
2
6 7
f x x x
( )− +
+
= − 2
2 2
3 7
3
6
x x
( )
= − − +
2
3 9 7
x
( )
= − −
2
3 2
x
6
3
2
=
General Form to Vertex Form
⚫ Probably the fastest method to find the vertex is to use
Desmos (surprise!) to graph the function. Desmos will
almost always automatically plot the vertex for you.
⚫ Using our same example:
The graph will show the
vertex as a gray dot.
General Form to Vertex Form
⚫ Probably the fastest method to find the vertex is to use
Desmos (surprise!) to graph the function. Desmos will
almost always automatically plot the vertex for you.
⚫ Using our same example:
The graph will show the
vertex as a gray dot. Click
to get the coordinates.
Practice
⚫ Find the vertex using the method of your choice. Then
give the axis of symmetry and the domain and range.
1. f(x) = x2 +8x + 5
2. g(x) = x2 – 5x + 8
3. h(x) = 3x2 + 12x – 5
Practice (cont.)
1. f(x) = x2 +8x + 5
The vertex is at (‒4, ‒11).
The axis of symmetry is x = –4.
The domain is (–∞, ∞), and the range is [–11, ∞)
( ) ( )
+
= + − +
2
2 2
8 4 4 5
f x x x
( )
= + − +
2
4 16 5
x
( )
= + −
2
4 11
x
( )
( ) ( )
2
8
4
2 1
4 8 4 5
16 32 5 11
h
k
= − = −
= − + − +
= − + = −
Practice (cont.)
2. g(x) = x2 – 5x + 8
The vertex is at , and the axis is .
The domain is (–∞, ∞), and the range is .
 
−
 
 
5 7
,
2 4
5
2
x = −
7
,
4
 


 
( )
2
5 5
2 1 2
5 5
5 8
2 2
25 25 25 50 32 7
8
4 2 4 4 4 4
h
k
−
= − =
   
= − +
   
   
= − + = − + =
Practice (cont.)
3. h(x) = –3x2 – 12x – 5
The vertex is at (‒2, 7), and the axis is x = –2.
The domain is (–∞, ∞), and the range is (–∞, 7].
( )
( ) ( )
2
12
2
2 3
3 2 12 2 5
12 24 5 7
h
k
−
= − = −
−
= − − − − −
= − + − =
Remember, the –3 flips the graph, so
the vertex is at the top.
Practice (cont.)
3. h(x) = –3x2 – 12x – 5 (Completing the Square method)
The vertex is at (‒2, 7), and the axis is x = –2.
The domain is (–∞, ∞), and the range is (–∞, 7].
( ) ( )
2
4 5
3
h x x x
= + −
−
( ) ( )
2 2 2
3
4 2 2
3 5
x x
= +
+
− + −
( )
2
3 2 12 5
x
= − + + −
( )
3 2 7
x
= − + +
Notice that I factored
the –3 out of the first
two expressions. This means that I’m
adding and
subtracting –3(22).
Practice (cont.)
3. h(x) = –3x2 – 12x – 5 (Desmos)
The vertex is at (‒2, 7), and the axis is x = –2.
The domain is (–∞, ∞), and the range is (–∞, 7].
Maximum and Minimum Values
⚫ The output of the quadratic function is the maximum or
minimum value of the function, depending on the
orientation of the parabola.
⚫ There are many real-world scenarios that involve
finding the maximum or minimum value of a quadratic
function.
⚫ If you are solving a problem involving quadratics,
and they ask for the maximum/minimum value, find
the vertex.
⚫ Depending on the problem, you will need either the
x-coordinate or the y-coordinate to solve the
problem.
Maximum and Minimum Values
⚫ Example: A backyard farmer wants to enclose a
rectangular space for a new garden within her fenced
backyard. She has purchased 80 feet of wire fencing to
enclose three sides, and she will use a section of the
backyard fence as the fourth side.
⚫ Find a formula for the area enclosed by the fence if
the sides of the fencing perpendicular to the existing
fence have length L.
⚫ What dimensions should she make her garden to
maximize the enclose area?
Maximum and Minimum Value
⚫ Example (cont.)
(We want to do this because the area will be A = LW.)
She only has 80 feet of
fencing available, so the
three sides would be
L + W + L = 80, or
2L + W = 80.
This means we can write W
in terms of L:
W = 80 – 2L
Maximum and Minimum Value
⚫ Example (cont.)
This is our quadratic function.
To find the maximum area, we find the vertex.
The area of the garden will
thus be
A = LW = L(80 – 2L)
or
2
2
80 2
2 80
A L L
L L
= −
= − +
Maximum and Minimum Value
⚫ Example (cont.):
( )
( ) ( )
2
80
20
2 2
2 20 80 20
800 1600 800
h
k
= − =
−
= − +
= − + =
( ) ( )( )
( )
2 2 2
2
2 40 20 2 20
2 20 800
20, 800
A L L
L
h k
= − − + − −
= − − +
= =
Maximum and Minimum Value
⚫ Example (cont.):
If you don’t see the parabola immediately, you may
have to zoom out and/or move the graph. If you still
don’t see it, make sure you typed the equation
correctly!
Maximum and Minimum Value
⚫ Example (cont.):
⚫ Whichever method you used, we have our vertex at
(20, 800). So what does it mean?
⚫ Remember our function was a function of L, which
was the length. So 20 represents the L at the
maximum area, which is 800.
⚫ To finish this out, the two perpendicular sides should
be 20 feet, the side between them will be
80 – 2(20) = 40 feet, and the maximum area for the
garden will thus be 800 square feet.
Classwork
⚫ College Algebra 2e
⚫ 5.1: 8-24 (×4); 4.3: 22-30 (even); 4.2: 46-58 (even)
⚫ 5.1 Classwork Check
⚫ Quiz 4.3

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5.1 Quadratic Functions

  • 1. 5.1 Quadratic Functions Chapter 5 Polynomial and Rational Functions
  • 2. Concepts and Objectives ⚫ Objectives for this section are ⚫ Recognize characteristics of parabolas. ⚫ Understand how the graph of a parabola is related to its quadratic function. ⚫ Determine a quadratic function’s minimum or maximum value. ⚫ Solve problems involving a quadratic function’s minimum or maximum value.
  • 3. Quadratic Functions ⚫ A function f is a quadratic function if where a, b, and c are real numbers, and a  0. ⚫ This is called the general form of the quadratic function. ⚫ The graph of a quadratic function is a parabola whose shape and position are determined by a, b, and c. ⚫ If a > 0, the parabola opens upward. ⚫ If a < 0, the parabola opens downward. ⚫ |a| determines the width of the parabola. ( )= + + 2 f x ax bx c
  • 4. Vertex Formula ⚫ We can use the general form to find the equation for the axis of symmetry and the vertex of the parabola: ⚫ If we use the quadratic formula, to solve for the zeros, we find the value of x halfway between them is always the equation for the axis of symmetry. ⚫ To find the y-coordinate of the vertex, plug x into the general form. 2 4 , 2 b b ac x a −  − = 2 0 ax bx c + + = , 2 b x a = −
  • 5. Vertex Form ⚫ The vertex (or standard) form of a quadratic function is written ⚫ The graph of this function is the same as that of g(x) translated h units horizontally and k units vertically. This means that the vertex of f is at (h, k) and the axis of symmetry is x = h. ⚫ The vertex form is just a vertical and horizontal shift of the parent function y = x2. ( ) ( ) 2 f x a x h k = − +
  • 6. Vertex Form (cont.) ⚫ Example: Graph the function and give its domain and range. ( ) ( ) 2 1 4 3 2 f x x = − − +
  • 7. Vertex Form (cont.) ⚫ Example: Graph the function and give its domain and range. Compare to : h = 4 and k = 3 (Notice the signs!) Vertex: (4, 3), axis of symmetry x = 4 We can graph this function by graphing the base function and then shifting it. ( ) ( ) 2 1 4 3 2 f x x = − − + ( ) ( ) 2 k f x a x h = − +
  • 8. Vertex Form (cont.) ⚫ Example, cont.: Let’s consider the graph of ⚫ Vertex is at (0, 0) ⚫ Passes through (2, ‒2) and (4, ‒8). ⚫ (I picked 2 and 4 because of the half.) ( )= − 2 1 2 g x x
  • 9. Vertex Form (cont.) ⚫ Example, cont.: To graph f, we just shift everything over 4 units to the right and 3 units up. Domain: (‒, ) Range: (‒, 3]
  • 10. Writing a Function From a Graph ⚫ Given a graph of a quadratic function, to write the equation of the function in general form: ⚫ Identify the vertex, (h, k). It will always be either the highest or lowest point of the parabola. ⚫ Substitute the values of h and k in the function ⚫ Substitute the values of any point, other than the vertex, on the parabola for x and f(x), and solve for a. ⚫ Expand and simplify to write in general form. ( ) ( ) 2 f x a x h k = − +
  • 11. Writing a Function From a Graph ⚫ Example: Write an equation for the quadratic function g in vertex form, and then in general form.
  • 12. Writing a Function From a Graph ⚫ Example: Write an equation for the quadratic function g in vertex form, and then in general form. First, we identify the vertex, which is at (–2, –3). Thus, h = –2 and k= –3, and our function looks like this: ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 3 2 3 g x a x g x a x = − + − = + −
  • 13. Writing a Function From a Graph ⚫ Example: Write an equation for the quadratic function g in vertex form, and then in general form. Now, we pick a point on the graph, for example, (0, –1), to substitute in for x and g(x). ( ) 2 1 0 2 3 4 2 1 2 a a a − = + − = =
  • 14. Writing a Function From a Graph ⚫ Example: Write an equation for the quadratic function g in vertex form, and then in general form. So, the vertex form for this graph is ( ) ( ) 2 1 2 3 2 g x x = + −
  • 15. Writing a Function From a Graph ⚫ Example: Write an equation for the quadratic function g in vertex form, and then in general form. To write this in general form, expand the binomial and simplify: ( ) ( ) ( ) 2 2 2 2 1 2 3 2 1 4 4 3 2 1 2 2 3 2 1 2 1 2 g x x x x x x x x = + − = + + − = + + − = + −
  • 16. General Form to Vertex Form ⚫ Given a quadratic function in general form, to find the vertex of the parabola (or to write in vertex form): ⚫ Identify a, b, and c from the general form. ⚫ Find h, the x-coordinate of the vertex, by substituting a and b into . ⚫ Find k, the y-coordinate of the vertex, by evaluating k . ⚫ Write in vertex form: 2 b h a = − ( ) 2 b f h f a   = = −     ( ) ( ) 2 f x a x h k = − +
  • 17. General Form to Vertex Form ⚫ Example: What is the vertex of the function? ( )= − + 2 6 7 f x x x
  • 18. General Form to Vertex Form ⚫ Example: What is the vertex of the function? From the general form, a = 1, b = –6, and c = 7. The vertex is at (3, ‒2). Vertex form is ( )= − + 2 6 7 f x x x ( ) 6 3 2 1 h − = − = ( ) ( ) 2 3 3 6 3 7 9 18 7 2 k f = = − + = − + = − ( ) ( ) 2 3 2 f x x = − −
  • 19. Completing the Square ⚫ Another method we can use is “completing the square” to transform it into vertex form (see section 2.5), but this time we do everything on the right-hand side. ⚫ For example, the function is not a binomial square. We can add 0 in the form of 52 – 52 (5 is half of 10), and group the parts that factor to a binomial square: ( ) + − = − + 2 2 2 10 30 5 5 f x x x ( )= − + 2 10 30 f x x x ( ) = − + − + 2 2 2 10 5 5 30 x x ( ) = − + 2 5 5 x
  • 20. Completing the Square (cont.) ⚫ Example: What is the vertex of the function? (same ex.) ( )= − + 2 6 7 f x x x
  • 21. Completing the Square (cont.) ⚫ Example: What is the vertex of the function? The vertex is at (3, ‒2). ( )= − + 2 6 7 f x x x ( )− + + = − 2 2 2 3 7 3 6 x x ( ) = − − + 2 3 9 7 x ( ) = − − 2 3 2 x 6 3 2 =
  • 22. General Form to Vertex Form ⚫ Probably the fastest method to find the vertex is to use Desmos (surprise!) to graph the function. Desmos will almost always automatically plot the vertex for you. ⚫ Using our same example: The graph will show the vertex as a gray dot.
  • 23. General Form to Vertex Form ⚫ Probably the fastest method to find the vertex is to use Desmos (surprise!) to graph the function. Desmos will almost always automatically plot the vertex for you. ⚫ Using our same example: The graph will show the vertex as a gray dot. Click to get the coordinates.
  • 24. Practice ⚫ Find the vertex using the method of your choice. Then give the axis of symmetry and the domain and range. 1. f(x) = x2 +8x + 5 2. g(x) = x2 – 5x + 8 3. h(x) = 3x2 + 12x – 5
  • 25. Practice (cont.) 1. f(x) = x2 +8x + 5 The vertex is at (‒4, ‒11). The axis of symmetry is x = –4. The domain is (–∞, ∞), and the range is [–11, ∞) ( ) ( ) + = + − + 2 2 2 8 4 4 5 f x x x ( ) = + − + 2 4 16 5 x ( ) = + − 2 4 11 x ( ) ( ) ( ) 2 8 4 2 1 4 8 4 5 16 32 5 11 h k = − = − = − + − + = − + = −
  • 26. Practice (cont.) 2. g(x) = x2 – 5x + 8 The vertex is at , and the axis is . The domain is (–∞, ∞), and the range is .   −     5 7 , 2 4 5 2 x = − 7 , 4       ( ) 2 5 5 2 1 2 5 5 5 8 2 2 25 25 25 50 32 7 8 4 2 4 4 4 4 h k − = − =     = − +         = − + = − + =
  • 27. Practice (cont.) 3. h(x) = –3x2 – 12x – 5 The vertex is at (‒2, 7), and the axis is x = –2. The domain is (–∞, ∞), and the range is (–∞, 7]. ( ) ( ) ( ) 2 12 2 2 3 3 2 12 2 5 12 24 5 7 h k − = − = − − = − − − − − = − + − = Remember, the –3 flips the graph, so the vertex is at the top.
  • 28. Practice (cont.) 3. h(x) = –3x2 – 12x – 5 (Completing the Square method) The vertex is at (‒2, 7), and the axis is x = –2. The domain is (–∞, ∞), and the range is (–∞, 7]. ( ) ( ) 2 4 5 3 h x x x = + − − ( ) ( ) 2 2 2 3 4 2 2 3 5 x x = + + − + − ( ) 2 3 2 12 5 x = − + + − ( ) 3 2 7 x = − + + Notice that I factored the –3 out of the first two expressions. This means that I’m adding and subtracting –3(22).
  • 29. Practice (cont.) 3. h(x) = –3x2 – 12x – 5 (Desmos) The vertex is at (‒2, 7), and the axis is x = –2. The domain is (–∞, ∞), and the range is (–∞, 7].
  • 30. Maximum and Minimum Values ⚫ The output of the quadratic function is the maximum or minimum value of the function, depending on the orientation of the parabola. ⚫ There are many real-world scenarios that involve finding the maximum or minimum value of a quadratic function. ⚫ If you are solving a problem involving quadratics, and they ask for the maximum/minimum value, find the vertex. ⚫ Depending on the problem, you will need either the x-coordinate or the y-coordinate to solve the problem.
  • 31. Maximum and Minimum Values ⚫ Example: A backyard farmer wants to enclose a rectangular space for a new garden within her fenced backyard. She has purchased 80 feet of wire fencing to enclose three sides, and she will use a section of the backyard fence as the fourth side. ⚫ Find a formula for the area enclosed by the fence if the sides of the fencing perpendicular to the existing fence have length L. ⚫ What dimensions should she make her garden to maximize the enclose area?
  • 32. Maximum and Minimum Value ⚫ Example (cont.) (We want to do this because the area will be A = LW.) She only has 80 feet of fencing available, so the three sides would be L + W + L = 80, or 2L + W = 80. This means we can write W in terms of L: W = 80 – 2L
  • 33. Maximum and Minimum Value ⚫ Example (cont.) This is our quadratic function. To find the maximum area, we find the vertex. The area of the garden will thus be A = LW = L(80 – 2L) or 2 2 80 2 2 80 A L L L L = − = − +
  • 34. Maximum and Minimum Value ⚫ Example (cont.): ( ) ( ) ( ) 2 80 20 2 2 2 20 80 20 800 1600 800 h k = − = − = − + = − + = ( ) ( )( ) ( ) 2 2 2 2 2 40 20 2 20 2 20 800 20, 800 A L L L h k = − − + − − = − − + = =
  • 35. Maximum and Minimum Value ⚫ Example (cont.): If you don’t see the parabola immediately, you may have to zoom out and/or move the graph. If you still don’t see it, make sure you typed the equation correctly!
  • 36. Maximum and Minimum Value ⚫ Example (cont.): ⚫ Whichever method you used, we have our vertex at (20, 800). So what does it mean? ⚫ Remember our function was a function of L, which was the length. So 20 represents the L at the maximum area, which is 800. ⚫ To finish this out, the two perpendicular sides should be 20 feet, the side between them will be 80 – 2(20) = 40 feet, and the maximum area for the garden will thus be 800 square feet.
  • 37. Classwork ⚫ College Algebra 2e ⚫ 5.1: 8-24 (×4); 4.3: 22-30 (even); 4.2: 46-58 (even) ⚫ 5.1 Classwork Check ⚫ Quiz 4.3