1. Quadratic Functions; An Introduction
Mr. J. Grossman

2. Graph the function y = x²
Describe the graph.
What does it look like?

3. Graph the function y = x²
The graph is…
Nonlinear
U-shaped smooth curve (called a parabola)
Symmetry
Axis of Symmetry: the line that divides the parabola into two matching halves.
It has a vertex.

4. Graph the function y = x²
The function we have graphed and described is known as a Quadratic Function.
A Quadratic Function is any function that can be written in the standard form:
y = ax² + bx + c
where a, b, and c are real numbers and a ≠ 0.

5. You must be able to recognize a Quadratic Function
From a graph…

6. You must be able to recognize a Quadratic Function
From a table of values…
Constant change in x-values x
y = x^2
1st Common Difference
2nd Common Difference 1 2 4 3 9 5 16 7

7. You must be able to recognize a Quadratic Function
A quadratic function will not have a common first difference. It will have a common second difference. This is true of all quadratic functions.
Constant change in x-values x
y = x^2
1st Common Difference
2nd Common Difference 1 2 4 3 9 5 16 7

8. You must be able to recognize a Quadratic Function
From an equation…
Must be a 2nd degree polynomial
y = 2x²
y = -5x² + 6x
f(x) = x² + 9
f(x) = ½ x² + 3x -11

9. Tell whether each function is a Quadratic Function:
{(-4, 8), (-2, 2), (0, 0), (2, 2), (4, 8)}
y = -3x + 20
y + 3x² = -4

10. Quadratic Functions
Some open upwards; others, downward.

11. Quadratic Functions
When a quadratic function is written in the form y = ax² + bx + c, the value of a determines the direction in which the parabola opens.
If a > 0, the parabola opens upward.
If a < 0, the parabola opens downward.
Remember, a ≠ 0 !!!

12. Quadratic Functions Graph the function: f(x) = -4x² - x + 1
Does the parabola open upward or downward?
Graph the function: y – 5x² = 2x – 6
Does the parabola open upward or downward?

13. Quadratic Functions
The highest or lowest point on the parabola is known as the vertex.
If a > 0, the parabola opens upward and the y-value of the vertex is the minimum value of the function.
If a < 0, the parabola opens downward and the y-value of the vertex is the maximum value of the function.

14. Quadratic Functions

15. Quadratic Functions
The value of a (coefficient of the x² term) affects the width of the parabola also.
The smaller the absolute value of a, the wider the parabola.
Compare the graphs of: y = -4x², y = ¼ x² + 3, and y = x².
Which graph (quadratic function) is widest? Narrowest?

16. Quadratic Functions
Recall that the quadratic function in standard form is written: y = ax² + bx + c.
The value of c (constant of the quadratic function) translates the graph of the function up or down the axis of symmetry.
Compare the graphs of the following quadratic functions: y = 2x², y = 2x² + 3, and y = 2x² - 3.

17. Quadratic Functions Positive values of c shift the vertex UP.
Negative values of c shift the vertex DOWN.

18. Quadratic Functions
DOMAIN: Unless a specific domain is given, you may assume that the domain of a quadratic function is the set of all real numbers.
RANGE: The range begins (albeit minimum or maximum value) with the vertex.

19. Any questions?
If not, complete Study Guide Practice 10-1, pg 128, all problems.