CHAPTER 6 Quadratic Functions

Transcription

1 CHAPTER 6 Quadratic Functions Math 1201: Linear Functions is the linear term 3 is the leading coefficient 4 is the constant term Math 2201: Quadratic Functions Math 3201: Cubic, Quartic, Quintic Functions Section 6.1: Exploring Quadratic Relations The path a ball travels gives a special U shape called a parabola.

2 Quadratic Functions: the shape is a parabola the simplest quadratic function is (The word quadratic comes from the word quadratum, a Latin word meaning square.) How to create a quadratic function? the result of multiplying two linear functions: Example: What do you notice about the degree (highest exponent of the variable) of the function?

3 Which of the following functions are quadratic? i) ii) iii) iv) v) vi)

4 Characteristics of the basic quadratic function. Create table of values x y y x What is the vertex? What is the x intercept? What is the y intercept? What is the domain and range? Can you visualize other types of parabolas?

5 Direction of Opening: parabola that can open up or down. When the graph opens up the vertex is the point on the graph and the y coordinate of the vertex is the value When the graph opens down the vertex is the point on the graph and the y coordinate of the vertex is the value.

6 Axis of Symmetry The parabola is symmetric about a vertical line called the axis of symmetry This lines divides the graph into two equal parts It is a mirror image It intersects the parabola at the vertex The equation of the axis of symmetry corresponds to the x coordinate of the vertex What is the equation of the axis of symmetry? Another Example: What is the equation of the axis of symmetry?

7 Relation vs Function Why are quadratic relations also quadratic functions? > For every value of x there is only one value for y. > It passes the vertical line test! Think about:

8 Standard Form of A Quadratic Function: where Terminology: = the quadratic term = the coefficient of the quadratic term Example: term and 3 is the term and 3 is the

9 Standard Form of A Quadratic Function: Investigate the parameters, and Part A: The Effect of a in on the graph of 1) What happens to the direction of the opening of the quadratic if or? 2) If the quadratic opens upward, is the vertex a maximum or minimum point? 3) If the quadratic opens downward, is the vertex a maximum or minimum point? 4) Is the shape of the parabola effected by the parameter? Are some graphs wider or narrower compared to the original? 5) What happens on the graph when?

10 Summary: The size of The sign of dictates the width of opening indicates direction of opening (i) If 1< a < 1, the parabola is wider compared to the graph y = x 2 (ii) If a > 1 or a < 1, the parabola is narrower compared to the graph y = x 2 (iii) If a is positive (a > 0) then the graph opens up (iv) If a is negative (a < 0) then the graph opens down

11 Part B. The Effect of on the graph of Compare graphs such as the following: What is the effect of parameter in on the graph of? b changes the location of the: and the

12 Part C. The Effect of c on the graph of Compare graphs such as the following: What is the effect of parameter in on the graph of? the c value changes the

13 SUMMARY: Standard Form y = ax 2 + bx + c Degree of all quadratic functions is 2 (largest exponent) A quadratic function is a parabola with a vertical line of symmetry The highest or lowest point, the vertex, lies on the axis of symmetry If a is positive, the graph opens up and if a is negative, the graph opens down If 1< a < 1, the parabola is wider compared to the graph of y = x 2 If a > 1 or a < 1, the parabola is narrower compared to the graph of y = x 2 Changing b changes the location of axis of symmetry and vertex Changing c changes the y intercept of the parabola

14 Complete the webbing below to describe the effects of the parameters a, b, and c on the quadratic function Practice: p. 324, #1, 2, 3, 5, 6