1. Chapter 10 Quadratic Equations and Functions
Section 10-1 Exploring Quadratic Graphs
A quadratic function is a function of degree 2
The graph of a quadratic function is a parabola
The line that cuts a parabola in half is the axis of symmetry
The highest or lowest point (depending on the direction of the parabola) is called the vertex → it is the point where the graph changes direction

2. Parabolas have reflectional symmetry over the axis of symmetry (they are exactly the same on both sides)
To graph a parabola: make a table of values until you have enough to plot and see the shape
The coefficient with the x² term determines the shape of the graph
If the coefficient is negative, the parabola is up-side-down (it has a maximum)
If the coefficient is positive, the parabola is right-side-up (it has a minimum)

3. If the absolute value of the coefficient is < n < 1, the shape will be wider than the parent function (y = x²)
If the absolute value of the coefficient is n > 1 then the shape will be skinnier than the parent function
Any number added to or subtracted from the x² term will shift the vertex
Quadratic formulas are used to model the height of a launched object over time

4. Graph
Ex1. y = 2x² + 4 Ex2. y = -¾x² – 5

5. Section 10-2 Quadratic Functions
The graph of y = ax² + bx + c, where a ≠ 0, has the line as its axis of symmetry
The x-coordinate of the vertex is
To graph a quadratic of the form y = ax² + bx + c:
A) find the axis of symmetry
B) find the vertex by plugging in your answer from part A in the equation for x (this will give you the coordinates of the vertex)

6. Graphing continued
C) Plug in other values for x and find the corresponding y-values (remember that since the graph has reflectional symmetry, the point equidistant on the opposite side of the axis of symmetry is also a point on the graph – see example 1 on page 518)
D) Connect
To graph a quadratic inequality, graph the parabola and then shade (< and < shade below, > and > shade above)

7. Ex1. y = 2x² + 4x – 3 A) Find the equation for the axis of symmetry
B) Find the coordinates of the vertex
C) Find 4 other points on the parabola
D) Graph

8. Ex2. Graph y > -x² + 6x – 5

9. Section 10-3 Finding and Estimating Square Roots
means the positive square root of x
means the negative square root of x
You should use the ± symbol to indicate the positive and negative answer
Ex1. Simplify
a) b) c) d)

10. Rational numbers end or repeat in decimal form
Irrational numbers do not end or repeat in decimal form (pi)
Ex2. List the first 15 perfect squares (know these!!)
Ex3. Between what two consecutive integers is ?
Ex4. Find the square roots of

11. Section 10-4 Solving Quadratic Equations
One way to solve a quadratic equation is by graphing
Graph the parabola and all of the places where the graph crosses the x-axis are the solutions (there could be one, none, or two)
If in the equation y = ax² + bx + c, b = 0, then you can solve by using square roots (simply solve for x)

12. Ex1. Solve by graphing x² – 9 = 0
Ex2. Solve by finding the square roots n² – 225 = 0

13. Review 10-1 through 10-4
10 – 1: Find the axis of symmetry and vertex for y = -6x² + 8
10 – 2: Find the axis of symmetry and vertex for y = 3x² + 6x – 5 (be able to graph too)
10 – 3: Simplify each of the following
a) b) c)
10 – 4: Solve. 81x² = 169

14. Section 10-5 Factoring to Solve Quadratic Equations
For every real number a and b, if ab = 0, then a = 0 or b = 0
By this property, you can solve to find the zeros (the solutions) of a quadratic formula by factoring
Factor and solve
Ex1. x² – 5x – 24 = 0
Ex2. x² – 9x = -20

15. Section 10-6 Completing the Square
Some of the other methods we have discussed for solve quadratics only works for certain types, completing the square works EVERY time
Completing the square involves adding the same number to both sides of an equation (which changes nothing) in order to make one side a perfect square trinomial

16. To complete the square, rewrite the equation so it is in the form x² + bx = c
Add (½b)² to both sides
Rewrite the left side as a binomial squared and simplify the right side
Then to solve, find the square root of both sides (two possibilities on the right) and solve for x
The directions should tell you when to round and when to give exact answers (if they don’t, give an exact answer)

17. Solve by completing the square.
Ex1. x² + 5x – 50 = 0
Ex2.
If there is a coefficient with the x² term, you will need to divide every coefficient in the equation by that number to eliminate it (the coefficient with x² must be 1)
Ex3. 3x² + 6x – 24 = 0

18. Section 10-7 Using the Quadratic Formula
The quadratic formula is another way to solve quadratic functions (it also works for all types)
x =
You must memorize this formula!
Notice the ± before the square root (this is because you get 1, 0, or 2 answers)

19. When using this formula, if you are going to put it all into your calculator at once, be VERY careful with parentheses! It is safer to put in small amounts at a time to avoid mistakes!
The radicand is what is underneath the radical sign (in this case b² – 4ac)
Read the purple box on the bottom of page 549 to know when to use which method of solving (when it is your choice)

20. Solve using the quadratic formula
Solve using the quadratic formula. Round to the nearest hundredth where necessary.
Ex1. x² + 2 = -3x
Ex2. 3x² + 4x – 8 = 0
Ex3. x² – 4x = 117

21. Section 10-8 Using the Discriminant
The radicand of the quadratic formula is called the discriminant (b² – 4ac)
The discriminant tells you how many answers there will be
If the discriminant is positive, there are two answers
If the discriminant is zero, there is one answer
If the discriminant is negative, there are no real answers

22. This is shown graphically on pages 554 and 555
This is shown algebraically in the orange box on page 555
Ex1. Determine the number of solutions to 6x² + 3x – 5 = 0
Remember that the number of solutions is also the number of x-intercepts (since the intercepts are the solutions)
Ex2. Determine the number of solutions to x² – 3x + 8 = 0

23. Section 10-9 Choosing a Linear, Quadratic, or Exponential Model
One way to determine which model is appropriate if you are only given a set of points is to graph those points and see what shape they make
There is another way that does not involve graphing
Open your book to page 560, we are going to see another way to determine

24. Once you have determined which model is best, you can use the equation for that model and the information given to write an equation for the situation
Which kind of data best models the data below? Write an equation to model the data.
Ex Ex2. x y 1
1.4 2
5.6 3
12.6 4
22.4 x y -1 4 2 1 .5 3
.25