1. Lesson 4

1.2. Explore

Mathematics 20-1 Module 3

Module 3: Quadratic Functions

 

Explore

 

In Lessons 1 and 2 you explored the effects of parameters a, p, and q in the vertex form. In Discover you investigated the standard form and the information you can obtain about the graphs of quadratic functions from parameters b and c. The following tables summarize the effect of all of these parameters on the graph.

 

 
Parameter Effect on Graph  
c

The y-intercept will always be the value of c in the function

y = ax2 + bx + c.

 This is the graph of y = 2x squared + 4x − 2. The y-intercept (0, −2) is marked with a red dot.

a

 This value influences the width of the curve of the parabola. This is the graph of y = 2x squared + 4x − 2 and y = 20x.
If a is negative, the parabola opens downward. This is the graph of y = −2x squared + 4x − 2.
If a is positive, the parabola opens upward. This is the graph of y = 2x squared + 4x − 2.
p The axis of symmetry is at x = p. This is the graph ofy = (x − 2)squared + 1 showing the parabolic graph, the equation, and the axis of symmetry at x = 2, which is the value of p.
q q is the minimum value if a > 0 or the maximum value if a < 0. The minimum or maximum occurs at y = q.

q is the minimum value if a > 0.

 

This is the graph of y = 2x2 + 4x – 2 and y = 2(x + 1)2 –4. The graph has an a value greater than 0. Therefore, the graph has a minimum value.

 

q is the maximum value if a < 0.

 

This is the graph of y = –2x2 + 4x – 2. The graph has an a-value less than 0. Therefore, the graph has a maximum value.
p and q The vertex is at (p, q).

This shows the graph of y = (x–2)2 + 1. The parabolic graph, the equation, and the vertex at (2, 1) are shown.


You may have noticed that parameter b was not included in the table. The effect of b on the graph is not an obvious pattern that will be studied in this course.

 

Try This 2

 

Begin with the quadratic function in standard form, y = 3x2 − 6x − 3.

  1. What information can you obtain by looking at the parameters a, b, and c?

  2. Graph this quadratic function using your graphing calculator or Quadratics in Polynomial Form - Activity B.

  3. Compare your answers from question 1 to the graph of the function you created in question 2. Do your predictions match what you see in the graph?

  4. Change the function from standard form to vertex form by completing the square.

  5. Graph the vertex form of this function using either your calculator or Quadratic Function (Vertex Form). How do these two graphs compare? What does this tell you?

  6. What other information can you find about the graph by looking at the parameters p and q in the vertex form? Do your predictions match your graphs?

    You may need to log in to LearnAlberta.ca first in order to get Quadratic Function (Vertex Form) to run.

     
    This is a play button that opens Quadratics in Polynomial Form - Actvity B.

    Screenshot reprinted with permission of ExploreLearning.

     
    This is a play button that opens Quadratic Function (Vertex Form).

course folder Save your response in your course folder.