1. Chapter 2 Functions and Graphs
Section 3
Quadratic Functions

2. Learning Objectives for Section 2.3 Quadratic Functions
The student will be able to identify and define quadratic functions, equations, and inequalities.
The student will be able to identify and use properties of quadratic functions and their graphs.
The student will be able to solve applications of quadratic functions.
Barnett/Ziegler/Byleen Business Calculus 12e

3. Quadratic Functions
If a, b, c are real numbers with a not equal to zero, then the function
is a quadratic function and its graph is a parabola.
Barnett/Ziegler/Byleen Business Calculus 12e

4. Vertex Form of the Quadratic Function
It is convenient to convert the general form of a quadratic equation
to what is known as the vertex form:
This is done by completing the square which will be reviewed in a few slides.
Barnett/Ziegler/Byleen Business Calculus 12e

5. Generalization For Vertex is at (h , k)
If a > 0, the graph opens upward.
If a < 0, the graph opens downward.
Axis of symmetry: x = h
k is the minimum if a > 0, otherwise its the maximum
Domain: −∞, ∞
Range:
If a < 0  −∞, 𝑘
If a > 0  𝑘, ∞
Barnett/Ziegler/Byleen Business Calculus 12e

6. Generalization
Barnett/Ziegler/Byleen Business Calculus 12e

7. General Form to Vertex Form Completing the Square
The example below illustrates the procedure:
Consider
Complete the square to find the vertex.
f (x) = (3x2 – 6x) – Group first two terms
f (x) = 3(x2 – 2x) –1 Factor out coef. of x2
f (x) = 3(x2 – 2x +1) –1 – Complete the square
inside the parentheses
Since you’re really adding 3, you have to subtract 3
f (x) = 3(x – 1)2 – 4 Vertex (1, -4); opens upwards
Axis of sym: x = 1; Minimum = -4
D: (-,  ); R: [-4, )
𝑓 𝑥 = 3𝑥 2 −6𝑥−1
Barnett/Ziegler/Byleen Business Calculus 12e

8. Example Rewrite the function in vertex form: 𝑓 𝑥 =− 𝑥 2 +8𝑥−9
𝑓 𝑥 =(− 𝑥 2 +8𝑥)−9
𝑓 𝑥 =−( 𝑥 2 −8𝑥)−9
𝑓 𝑥 =− 𝑥 2 −8𝑥+16 −9+16
𝑓 𝑥 =− 𝑥−
Vertex (4, 7); opens downwards
Axis of sym: x = 4; Maximum = 7
D: (-,  ); R: (-, 7]
Barnett/Ziegler/Byleen Business Calculus 12e

9. Intercepts Y-intercept Plug in x = 0
Barnett/Ziegler/Byleen Business Calculus 12e

10. Intercepts y−intercept is:−1 Find the y intercept of:
Barnett/Ziegler/Byleen Business Calculus 12e

11. Intercepts X-intercepts It might have 0, 1, or 2 x-intercepts
They can be determined by:
Factoring (if possible)
Completing the square
Quadratic Formula
Barnett/Ziegler/Byleen Business Calculus 12e

12. Intercepts Find the x intercepts of 𝑓 𝑥 = 𝑥 2 +5𝑥−14 0=(𝑥+7)(𝑥−2)
𝑥+7=0
𝑥−2=0
𝑥=−7
𝑥=2
The x−intercepts are:−7 and 2.
Barnett/Ziegler/Byleen Business Calculus 12e

13. Intercepts Find the x intercepts of Using the quadratic formula:
𝑥= −6± −4(−3)(−1) 2(−3)
= −6± 24 −6
Exact
Approx.
= −6±2 6 −6
= −3± 6 −3
≈1.82, 0.18
Barnett/Ziegler/Byleen Business Calculus 12e

14. Finding x-intercepts Using a Graphing Calculator
Graph: y = –x2 + 5x + 3
Select CALC (2nd Trace)
Select 2: zero
Left bound? Use arrows to position cursor to the left of intercept, then hit ENTER.
Right bound? Use arrows to position cursor to the right of intercept, then hit ENTER.
Guess? Hit ENTER.
Zero 
Repeat to find other zero.
Zero 
Barnett/Ziegler/Byleen Business Calculus 12e

15. Max and Min Values A parabola that opens upwards has a minimum value.
A parabola that opens downwards has a maximum value.
In either case, the max/min value is the y-coordinate of the vertex.
Finding the vertex from the equation:
𝑓 𝑥 =𝑎 (𝑥−ℎ) 2 +𝑘  Vertex (h, k)
𝑓 𝑥 = 𝑎𝑥 2 +𝑏𝑥+𝑐  Vertex −𝑏 2𝑎 , 𝑓 −𝑏 2𝑎
Barnett/Ziegler/Byleen Business Calculus 12e

16. Example Find the maximum or minimum of each function:
𝑓 𝑥 = −(𝑥+3) 2 +7
𝑓 𝑥 = 3𝑥 2 −12𝑥+14
− 𝑏 2𝑎 =− −12 6 =2
𝑉𝑒𝑟𝑡𝑒𝑥 (−3, 7)
𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑜𝑓 7.
𝑓 2 =2
𝑉𝑒𝑟𝑡𝑒𝑥 (2, 2)
𝑀𝑖𝑛𝑖𝑚𝑢𝑚 𝑜𝑓 2
Barnett/Ziegler/Byleen Business Calculus 12e

17. Finding Max/Min Using Graphing Calculator
Graph: y = –x2 + 5x + 3
Select CALC (2nd Trace)
Select 4: maximum
Left bound?
Right bound?
Guess?
Maximum 9.25
Vertex: (2.5, 9.25)
Barnett/Ziegler/Byleen Business Calculus 12e

18. Quadratic Inequalities
Two Methods for Solving Quadratic Inequalities
Algebraic
Do not need to review this procedure yet.
Graphing Calculator
This is the procedure we will use for now.
Barnett/Ziegler/Byleen Business Calculus 12e

19. Quadratic Inequalities
Graph the function.
Determine its x-intercepts using the Calc Zero function.
If inequality is:
f(x) > 0 then state the intervals for which the graph is above the x-axis.
f(x) < 0 then state the intervals for which the graph is below the x-axis.
f(x)  0 then state the intervals for which the graph is on or above the x-axis.
f(x)  0 then state the intervals for which the graph is on or below the x-axis.
Barnett/Ziegler/Byleen Business Calculus 12e

20. Solving Quadratic Inequalities
Solve the quadratic inequality –x2 + 5x + 3 > 0 .
x-intercepts are: and
The graph is on or above the
x-axis over the interval:
[– , ]
Barnett/Ziegler/Byleen Business Calculus 12e

21. Solving Quadratic Inequalities
Solve the quadratic inequality –x2 + 5x + 3 < 0 .
x-intercepts are: and
The graph is below the x-axis over the interval:
(−∞, – )  ( , ∞)
Barnett/Ziegler/Byleen Business Calculus 12e

22. Applications
There are many applications involving quadratic functions.
Let’s look at an example…
Barnett/Ziegler/Byleen Business Calculus 12e

23. Break-Even Analysis
The financial department of a company that produces digital cameras has the revenue and cost functions for x million cameras as follows:
R(x) = x(94.8 – 5x)
C(x) = x. Both have domain 1 < x < 15
Break-even points are the production levels at which R(x) = C(x). Find the break-even points (using your graphing calculator) to the nearest million cameras.
Barnett/Ziegler/Byleen Business Calculus 12e

24. Solution to Break-Even Problem (continued)
If we graph the cost and revenue functions on a graphing utility, we obtain the following graphs, showing the two intersection points:
x = or
The company breaks even when they sell approximately 2 million cameras and 13 million cameras.
Barnett/Ziegler/Byleen Business Calculus 12e

25. Quadratic Regression Insert speed in L1 and mpg in L2. Turn Plot 1 on
Evinrude Outboard Motor
Insert speed in L1 and mpg in L2.
Turn Plot 1 on
ZoomStat
STAT
CALC
5:QuadReg
Regression equation is:
𝑦=− 𝑥 𝑥−0.1794
Speed (mph)
Miles per Gallon
10.3
4.1
18.3
5.6
24.6
6.6
29.1
6.4
33.0
6.1
36.0
5.4
38.9
4.9
Barnett/Ziegler/Byleen Business Calculus 12e