Quadratic Functions

1. Section 1.4
Quadratic Functions

2. Parameters
• Parameters are numbers that affect how a
mathematical function operates or how its
graph looks.
• In a linear equation you have parameters
‘a’ and ‘b’:
• y = f(x) = ax + b
• ‘a’ is ROC/ slope = Δy = y2-y1
Δx x2-x1
‘b’ is the y intercept or initial value.
E.g y = 2x + 1
Slope is 2 and the y intercept is 1.

3. Parabolas
• The graph of the quadratic function is
called a parabola.
• The 3 forms of the quadratic function are:
– General: y = ax2 + bx +c
– Standard: y = a(x-h)2 + k
– Factored or Zero: y = a(x-z1)(x-z2)
– Where a, b, c, h, k, z are numbers

4. Basic Quadratic Function Properties
Rule y = 1x2
Graph Parabola, opens up
Vertex at (0,0)
Domain ]-∞, ∞ [
Range [0, ∞ [
Extreme(s) (if any) Minimum of 0 (y value)
Zero At x = 0
Sign Strictly positive
Variation Decreases ]-∞,0]
Increases [0, ∞ [
Is the inverse a function No, fails vertical line test

5. Transformed Quadratic Function Properties: General
Rule y = ax2 + bx + c
Graph If a>0, opens up
If a<0, opens down
C – vertical translation
Domain ]-∞, ∞ [
Range depends
Extreme(s) (if any) If a > 0, minimum
If a < 0, maximum
Zero x = -b ±√(b2 – 4ac)
2a
Sign Depends
Variation Depends

6. Transformed Quadratic Function Properties: Standard
Rule y = a(x-h)2 + k
Graph a – same as in general
Vertex at (h,k)
Domain ]-∞, ∞ [
Range depends
Extreme(s) (if any) Same as in general
Zero x = h ± √ - k
2a
Sign Depends
Variation Depends
Is the inverse a function No, fails vertical line test

7. Transformed Quadratic Function Properties: Zero
Rule y = a(x-z1)(x-z2)
Graph a – same as in general
Zeroes at x =-z1 & -z2
Domain ]-∞, ∞ [
Range depends
Extreme(s) (if any) Same as in general
Zero Zeroes at x =-z1 & -z2
Sign Depends
Variation Depends
Is the inverse a function No, fails vertical line test

8. Parameters: a
• Parameter ‘a’ appears
at the beginning of a
function.
• It has the affect of
multiply by ‘y’ value
of the function.
• This acts as a vertical
scale change
x y = x2 y = 3x2
1 1 3x1 =
3
2 4 3x4 =
12
-4 16 3x16=
48

9. Parameters: h,k
• Parameter ‘h’ appears
inside the bracket of a
function and are added to
the ‘x’.
• Parameter ‘k’ appears
outside the bracket and is
added to the ‘y’.
• h&k act as horizontal and
vertical translations
respectively.
y= x2
x y
y=(x-2)2 +3
x y
1 12
= 1
1 + 2
= 3
1 + 3
= 4
2 22
= 4
2 + 2
= 4
2 + 3
= 5
3 32
= 9
3 + 2
= 5
3 + 3
= 6

10. Axis of Symmetry
• A parabola has a vertical axis of symmetry
at x = h
• Each side is a mirror image of the other
• You can use this to find zeroes and other
points

11. Activities
• Page 93, Q. 1-3, 5,6
• Page 96, Q. 10 -13
• Page 98, Q. 14 - 17

12. Square Root Function
• Basic Equation:
• y = √x
• Transformed
Equation: y = a√bx
• E.g. y = 2 √4x
• y = 2*2√x
• y = 4√x
x y = √x y = 2√4x
y = 2*2√x
0 0 4*0 = 0
1 1 4*1 = 4
4 2 4*2 = 8
9 3 4*3 = 12

13. Determining the Vertex
• Using the general formula the vertex (h,k)
is at
• h = -b

14. Square Root Function
• Basic Equation:
• y = √x
• Transformed
Equation: y = a√bx
• E.g. y = 2 √-4x
• y = 2*2√-x
• y = 4√-x
x y = √x y = 2√-4x
y = 2*2√-x
0 0 4*0 = 0
1 1 4*1 = -4
4 2 4*2 = -8
9 3 4*3 = -12

15. Determining the Equation
• Basic Equation:
• y = √x
• Transformed
Equation: y = a√bx
• Substitute
• 10 = a√4
• 10 = a * 2
• 10/2 = 5 = a
• So y = 5√x
x y = √x y = a√bx?
0 0 0
1 1 5
4 2 10
9 3 15

16. Solving a Square Root Equation
• You may be asked to determine when the
equation from the last slide = 40
• y = 5√x
• 40 = 5√x
• Divide both sides by 5
• 8 = √x
• x = 2.8
• So when x = 2.8, y = 40

17. Inequalities and
Square Root Functions
• To solve inequalities, we treat them like
equations.
• When is 5√x < 50?
• Divide both sides by 5
• √x < 10
• Square both sides to get rid of square root
• x < 100
• The point x would NOT be part of the solution.

18. Quadratic or 2nd Degree Function
• Basic Equation:
• y = x2
• Transformed
Equation: y = ax2
• E.g. y = 2 (3x)2
• y = 2 * 9x2
• y = 18x2
x y = x2 y = 18x2
0 0 18*0 = 0
1 1 18*1 = 18
3 9 18*9 = 162
5 25 18*25 = 900

19. Determining the Quadratic Equation
• Basic Equation:
• y = x2
• Transformed
Equation: y = ax2
• Substitute
• 28 = a *22
• 28 = a * 4
• 28/4 = 7 = a
• So y = 7x2
x y = x2 y = ax2
0 0 0
1 1 7
2 4 28
3 9 63

20. Solving a Quadratic Equation
• You may be asked to determine when the
equation from the last slide = 400
• y = 18x2
• 400 = 18x2
• Divide both sides by 18
• 22.22 = x2
• x = ±4.71, positive and negative!
• So when x =± 4.71, y = 400

21. Inequalities and
Quadratic Functions
• To graph inequalities, we treat them like
equations.
• y ≥ 7 x2
• We would draw the graph y = 7 x2
• Because the equation is greater than AND
equal to, we shade above the line and
make the line SOLID.

22. Inequalities and Parabolas 1
• Pick a point
• Sub in.
• Is it true or not?

23. Substituting
• y > x2
• Sub a point (0,1) in
and see if the
mathematical
equation is true or
not.
y > x2
1 (0)2
1 0
True, therefore
(0,1) is in the
region y>x2

24. Inequalities and Parabolas 2
• Pick a point
• Sub in.
• Is it true or not?

25. Substituting
• y < x2
• Sub a point (2,0) in
and see if the
mathematical
equation is true or
not.
y < x2
0 (2)2
0 4
True, therefore
(2,0) is in the
region y<x2

26. Activity
• Page 43
• Questions 3, 5, 9, 10 ,12