Module 2: QUADRATIC FUNCTIONS
Exploring Calculus

There are many important phenomena in science, engineering and business which are more complex than those which can be described by linear functions. Many of these situations must be described by non-linear functions, such as quadratic functions. Quadratic functions have a maximum or minimum value, which you will see is of major importance in solving application problems.

Critical Thinking Questions

1. We will work with the graphs of three quadratic functions. Initially these are     

    y₁ = x²
    y₂ = - x²
    y₃ = x² + 5

    a. How are the graphs of y₁ and y₂ different? What part of the equation do you think makes them different?

    b. How are the graphs of y₁ and y₃ different? What part of the equation do you think makes them different?

    c. Each graph above has a maximum or a minimum point. What is it?

2. Now, let     

    y₁ = x² + 3
    y₂ = x² + 4x + 3
    y₃ = x² - 2x + 6

    a. What is the minimum point on each graph called? What are the points in each case?

    b. How do y₂ and y₃ behave differently than y₁?

    c. Which part of the equation in y₂ and y₃ makes them behave differently than y₁?

    d. Why does a quadratic expression of the type in y₂ and y₃ have a vertex off the y-axis?

3. The following questions will help you determine how the coefficient of the x² term affects the graph of a parabola.

    a. Describe how the graphs of the parabolas y₁ = x² , y₂ = x² , and y₃ = 4x² are different.
                          Which part of the equation makes them different?

    b. Describe how the graphs of y₁ = 4 - x² and y₂ = x² + 4 are similar and how they are different.
                          Which part of the equation makes them different?

4. Given the general quadratic function y = ax² + bx + c.

    a. What algebraic procedure is used to rewrite it so that its vertex is displayed?    Hint

    b. How do you find its x- and y-intercepts?

    c. What are its x- and y-intercepts?

5. Let Q(x) be any quadratic function.

    a. It will always cross the __- axis _____ time(s).

Hint: There is only one correct answer.

    b. What are all the possible number of x-intercepts Q(x) may have?

    c. Sketch a freehand graph illustrating each case in Q.5b.

6. When a quadratic is written in the form y = a (x - h)² + k

    a. Name the point which is easy to read from the equation.

    b. What are the coordinates of that point?

    c. What are all the ways the value of a can affect the graph of a quadratic function?

Skill Exercises

1. Consider the parabola y = 2(x + 5)² - 2.

    a. What is its vertex?

    b. Which way does the graph open?

2. For each of the three parabolas

    y₁ = x² + 5
    y₂ = x² - x - 6
    y₃ = -2x² + 4x + 3

find

a. the vertex

b. the x- and y-intercepts

Show the algebra used to obtain your answer and verify it graphically.

Summarize your results in the table below.

.	y1	y2	y3
Vertex	........	........	........
x-intercept(s)	.	.	.
y-intercept	.	..	.

3. Let y = -2x² + 3x + 4 .

    a. Which way does the parabola open?

    b. Is its vertex on the y-axis?    Verify your answer algebraically.

4. Let y = -x² + 6x + 2.

    a. Find the maximum point on the graph.

    b. Graph the parabola and draw a line tangent to the graph at the maximum point.

    c. What is the slope of this line?

5. If an object is thrown upward with an initial velocity of 32 feet per second, then its height, h(t) after t  seconds is given by

h(t) = 32t - 16t²

Verify your answers algebraically.

    a. What is the maximum height attained by the object?

    b. How many seconds does it take for the object to hit the ground?

Exploring Calculus